If your piece lands on the tile currently occupied by the Time Breaker...

...you can apprehend them. But you have to declare that they are under arrest. Assuming you have done so, they will move along with you whenever your piece moves thereafter. It’s considered good form to make the declaration as soon as you land on the Time Breaker, but as long as you say it before you try to move again, they will accompany you.

If another player lands on the same space...

...they must also declare the Time Breaker to be under arrest. The Time Breaker will move with whichever player leaves first, regardless of who arrested them first.