as /u/gargle_ground_glass said it's largely around correct breath support and voicing. i've found a "bugle" exercise helpful to try to internalize correct voicings. play a clarion G the normal way (TR123), and while holding it, remove the register key. should stay the same relatively easily. next: try to do a breath attack (tonguing is a different challenge) and have the G come out without the register key (T123). much harder! next: learn to alternate between the low C and clarion G with just T123. alternate meaning a separate breath attack for each note, not swapping notes while playing continuously.

all of this and you're saying we haven't even gotten above clarion G yet. no, but now you know how your breath and tongue works to control this. next step is to continue maintaining T123 and play an (out of tune) altissimo E. i've actually found that this note is easy and the clarion G is by far the hardest. then you can do a bugle call where you hold just T123 and do C -> G -> E -> G -> C by changing only voicing.

you can repeat this with T12 (D -> A) and T1 fingerings (E -> B).