Is this right?
 

College game--

A1 throw-in frontcourt endline...

Throw-in is towards sideline, ball hits floor and bounces high and is headed OOB. A2, in frontcourt, runs to ball, jumps (both feet in air) then catches ball (definite 2-hand possession), throws ball to backcourt and then lands OOB. A3 goes to backcourt and retrieves. No whistle.

Is this right?

Backcourt Violation. Someone else will probably provide a rules reference.
Essentially, Possession by A2 ends the throw-in and the throw-in exception. Now you have A2 with FC status and team control. Ball thrown into the BC,
A3 first to touch. Backcourt violation.

As long as it was clear A2 possessed the ball, BC violation. If it was unclear and could have been a bat, no violation.