I have often observed brass playing which seems to follow a methodology and mindset that often creates some fairly poor results.
One of the comments that seems to be widespread is that if the music asks the performer to play forte (loudly), it is an invitation to play at the loudest possible volume simply by blowing as hard as possible. With this kind of approach, what is normally heard is a fairly uncontrolled explosion of noise. If the noise that comes out is actually a good enough sound to be a note, it often goes quite sharp in pitch. Thinking in this way often leads toward the belief that you play softly by not blowing much air, leading to a weak sound with little support that goes flat in pitch.
In line with this issue is that when the dynamic changes from soft to loud, the speed sometimes increases with the change in dynamics. Obviously having speeds fluctuate with dynamic contrast produces fairly unmusical results and makes the music feel as those it rocks in terms of ensemble.
When these issues exist it is possible to work around them with some focused practice as well as a small adjustment in technique and perception.
How can these issues be fixed?
The first issue to think about is how a controlled loud or quiet sound is produced. It is important to remember that producing a loud or soft note requires the use and support of air. Replace the thinking that loud sounds are produced by blowing hard, by thinking about the speed of the air instead.
By thinking that the way to play loudly is by moving lots of air quickly through the instrument you will get a cleaner production with a fuller, warmer and more controlled sound. To produce a quiet sound the best way of getting a supported and full sound is to think of moving lots of air slowly through the instrument. This will allow the embouchure the opportunity to work properly and for the sound that is produced to be well supported with good tonal quality.
Therefore, whether playing loudly or softly it is crucial to use lots of air. The only variation should be the speed at which the air is moved. With practice, this will lead to a greater control and awareness of how to produce a much wider range of dynamics with good tonal quality.
This approach will normally rectify any issues with tuning when playing louder or more softly as the supported note will be far more controlled. It is of course very important to continue to listen as carefully as possible to intonation as there will inevitably be variation across different ranges of the instrument. Practice within ensembles, as well as careful work on scales and arpeggios will also help to improve intonation across the whole range.
Using a metronome in practice will help will fluctuation in tempo if practiced consistently over time. This will develop a reliable internal beat that should work well in conjunction with the change in perception of how to change dynamic levels more effectively.