When you produce a tone with your instrument, you are not only using your lips but also all the muscles involved in breathing. You need to make sure that you coordinate all these muscles with your lips properly to get the best possible result. 

One comment we hear about a singer sometimes is that "His tone really comes from deep". That's because they need to use their entire torso to move the air through their vocal cords to produce a nice tone. We can learn from that and learn to support our chops with a good air stream. 

It's not very complicated after all. Here is something you can do to practice your air support:

1) Stand up or sit with your back straight.

2) Fill your lungs with air completely and let the air come out without making any effort to push it out. When you breath, extend your entire rib cage as much as possible. 

3) After a few breaths, place your hand in front of your mouth and blow in it. Try to maintain the same air pressure for as long as possible. You will have to apply gentle pressure from your rib cage and diaphragm to keep the air pressure. Don't blow to hard, It's not necessary.
Do this a few time and be aware the muscles you are using to keep the air running. 

4) Repeat the exercise using your mouthpiece this time. Hold long notes. You can try various pitches. You'll notice that it gets harder to stay relaxed as you get higher. Try to keep your torso muscles from getting tense as you go higher.

5) Now put your mouthpiece on your instrument and do the same thing with various notes. Once you feel comfortable, try scales and arpeggios. 

Try to be aware that you're are using your entire body to produce your tone. Not only your chops. Feel which muscles are working.
You will have more power in the high register and more endurance if your lips are well supported by a good air stream that comes from the bottom. 

You can improve your tone a lot by playing your music at a very slow tempo. 
First, try to imagine the most beautiful sound in your head and then play the first note of your piece while still mentally producing that tone. After a while, your brain will send signals to the rest of your body that this is the tone it wants and your tone will adapt. Once you're happy with the sound, do the same with the next few notes. Remember to always focus on the tone you are making in your head. 

It's a bit like doing tai chi on your instrument. You do everything in super slow motion. You don't need to play the whole piece like that. Just the first few passages. After a short while, your tone will match your mental projection quicker and you won't need to play so slow to get what you want. This will relax you and give you a nice smooth tone. 

Everyone gets a bit stiff once in a while. We spend a lot of time practicing difficult passages and pieces and that can sometimes create tensions affecting the embouchure and making our playing uncomfortable especially in the high register. 

If your embouchure doesn't feel as good as it normally does, you better not insist too much and avoid forcing the notes to come out. It will only make things worst. If you can normally play a piece well, there is no need to fight with it if it doesn't work on that day. 
Instead of trying to beat your instrument into submission, just put the hard stuff away and play some very simple, easy and slow pieces. Even beginner pieces. Stop on a few notes and try to make them sound as good as possible without forcing. After about 30 minutes of doing this, you'll feel your embouchure relaxing and feeling right again.
After that, take a break or put the horn away for the day.  When you pick it up again, go back to what you normally practice. You should feel much better.  

It is important to take a break from your instrument sometimes. I just came back from a couple of weeks of vacation myself! Not playing for a little while will help your body to get rid of tensions accumulated during the past few months and rest your head while you think of something else. When you come back to your instrument,
you're more enthusiastic about playing and ready to rock!

The downside of this is that you'll be out of shape when you pick up your horn again. If I don't do anything during my break, it normally takes me one day of practice for each day of break before I get my optimal shape back. For example, it would take me two weeks to get back in full shape after a two weeks rest. That's why I always plan to come back from vacation several days before we start working again to get my chops running again. If I can't do that, I'll just bring my horn with me and play during my trip. 

There are things you can do to speed up the process. I did the pencil exercise every day for a week before coming back and it helped maintaining some strength in my muscles. I also buzzed my mouthpiece a bit for 15 minutes/day during my trip. Once I got back to my horn, I did the Caruso technique every second day to strengthen my embouchure. Doing all this cut my recovery time in half and I'm almost ready for work after 5-6 days.

You need to be patient when you come back from a break. The first practice session will most likely not be very pleasant. Especially if you add jet lag on top of that! Increase the duration of your practice sessions and level of difficulty of what you play gradually. At first, play some very easy pieces and just get through them. The next day, try to play a bit longer and work on your tone a bit. When you feel a bit stronger, start playing etudes. They are very good for endurance since they normally have no or very few bars of rest so your chops can get use to play non stop for a while again. Like I said, the Caruso technique or any other chops building method is always good. With all this, you should be able to get back in shape in time for your next gig with a strong embouchure, a well rested mind and a relaxed body!

Buzzing with your lips alone without instrument or mouthpiece will improve your tone and strengthen your embouchure. It may also solve some embouchure position problems.

To get a nice buzz, bring your upper lip slightly on top of the lower one and blow some air. With a bit of practice, you can control the pitch of your buzz. 

I often practice buzzing a few simple exercises ( just a couple of intervals) which I repeat on my mouthpiece and then on the horn. The tone is instantly richer and more focused. 

Here is a little example. It's very simple. 

Many musicians, actors, dancers and athletes take Alexander technique lessons to improve the way they use their body. I have personally done it myself for about a year in Switzerland back in 2001-02. AT is not only about having the right posture when you play but also about making the right move with your entire body to achieve an optimal result. It teaches you not to focus on the actual results but on the means to get there. If you do the right move, you'll eventually get the right result with time and patience.

We very often take shortcuts to reach notes or play a difficult passage. Although you might actually get what you want in the short term, you can easily develop bad habits from forcing things which will hurt you in the long run. AT can help you unlearn some of these bad habits and replace them with good ones. 

I was always amazed how good my playing felt after each Alexander technique lesson. Some conservatories and music faculties offer it on their premises. If you are lucky enough to be in one of these schools, take advantage of this. If your school doesn't have an AT teacher, make a little research to find one in your area. You won"t regret it. You'll immediately feel a difference in your tone and it'll greatly improve the fluidity of your playing. 

Here is a video of an Alexander technique masterclass to give you an idea of what it can do for you. You can find many more about it online. 

Here's a little experiment: make a fist with both hands and squeeze as hard as possible. While doing that, try to blow air. How does it feel? Not very good of course.

Now relax both hand and blow. See how natural and comfortable that feels. 

We tend to press our valves or pistons too hard when we play. Especially for high or difficult passages. This creates tension in the harm which affects the rest of the body and stiffens the air flow causing problems in the high register. 

Take a couple of minutes to practice pressing your valves gently with just the minimum amount of pressure. Trombone players can practice holding the slide gently and moving the harm fluidly. Then play on your instrument this way. Your playing should immediately feel more relaxed and comfortable. You might miss a bit more notes at first but it'll only take a few minutes for your embouchure to get used to this new relaxed way of playing. 
This will help your tone in the high register and improve your endurance. 

In many ways brass playing is similar to singing. Once you have conquered the technical abilities to play concertos, etudes and exerps, what you will be remembered for is your tone and musicality. What matters is not WHAT you play but HOW you play it. We all remember Luciano Pavarotti for his unforgettable  voice. Have you ever heard anyone say "Man! That guy Pavarotti can sing so fast!" ? Of course not! It's his tone that everyone remembers. 

There are many things you can do to improve your tone. Here are a few ideas:

1) Long tones. Play 8 beats crescendo and 8 beats diminuendo on a chosen note. Try to maintain a nice warm sound through the whole note. Repeat as many times as you want with other notes. 

2) Practice easy pieces and make them sound as good as possible. We are so busy practicing high, fast and loud stuff that we forget to practice soft, slow middle range pieces. Most solos in the orchestral repertoire aren't that difficult technically but require great lyrical skills. Also, playing easy pieces is relaxing and makes you embouchure feel good again if your chops are stiff. 

3) Hear the tone that you want in your head. Take a few seconds to imagine the greatest sound in your head before you play. Then, do a few long tones while keep producing that tone in your brain. Your brain will  send signals to the rest of your body to make it happen. After a few minutes doing this, you should see your tone changing. 
Try to play your pieces, etudes and exerps slowly using THAT tone that you just made and speed up to the desired tempo when you feel comfortable. 

4) Stay in shape...

5) But don't over practice. Use your judgement to determine how much you really need to practice. It depends on your level, what you need to play, what are your goals etc. It's hard to sound good if you rarely touch your instrument. On the other hand, nothing good comes out of practicing on worn out chops. Practice a lot but also give your chops some rest.

6) Don't over blow. Unless the conductor asks for more, play at a dynamic you can control. Of course, in your practice room, you will work to expand your dynamic range while keeping a nice sound so you'll have it when you need it. Needlessly forcing air into your horn will only give you a harsh tone without that much more power.

7) Don't be too perfectionist. I never play well when I try to be perfect. It makes me tense, it's boring and I still miss notes! Let yourself be human and you'll sound a lot better.

8)Try different mouthpieces. Take all the mouthpieces at a music store or at your school and try them all. See which one feels and sounds the best. A different mouthpiece can change your tone a lot. 

9) Think about it when you play. When playing fast passages,we are often so busy playing the notes that we forget about the tone. Sometimes, just reminding yourself to produce a nice sound will solve the problem. 

I'm sure there are many other things that can improve your tone. This is what I can think of for now. If you have other ideas, write them in the comment section and I'll add it to the list.

It is important to have a relaxed and fluid air flow while you play. I had the chance to have a few lessons with Mr. Vincent Chicowitz (former principal trumpet of the Chicago symphony) who taught me to practice just the air movement without any instrument or mouthpiece. Basically, you just practice your music blowing the rhythm as you would blow in your instrument if you were playing normally. Groups of slurred notes are blown like one long note like a tie over. You practice a difficult passage like this two or three times and then you go back to your instrument. It feels great! It really helps for passages in the high register. I personally do it every time I feel my air is a bit stiff. 

I personally practice on my mouthpiece alone every day for a few minutes as part of my warm up routine. I do  one or two octaves glissandos to increase flexibility and fluidity.  It is also very useful to play a difficult passage on the mouthpiece and then play it on the horn. It allows you to focus on the actual movement of your lips and air flow. It's also a good way to find out if you hear the music correctly.