How to develop a relaxed and lyrical high register by using the Vocalises by Marco Bordogni transcribed by Joannes Rochut.
Play the vocalise as if it were written in tenor clef down two octaves. You do this by ignoring the bass clef and imagine there is a tenor clef instead. In regards to the key signature, you will transpose up one 5th. For example a vocalise in C major will now have a the signature of G major.
Play down two octaves with a rich full sound. Breath often but try to still make music. Start by just playing one phrase or one line. Do this 3 or 4 times. Now buzz the vocalise on the mouthpiece this time as if the study were in tenor clef. Again just start with one phrase or line. Do this 2 or 2 times. Now play the vocalise in tenor clef on the trombone and try to emulate the same free blowing feeling you had in the low register. Take this approach for all lyrical passages in the high register.
Here is a video from my trombone camp where I work with a student on this approach.