FW here, but I've done ground EMS and think it's similar for this question.
A lot depends on the anticipated length of the transport. If it means we'll run over-shift by an hour, two or even three - the present crew will take it. (That's why we shouldn't plan anything important for the morning after the 24 hour shift.) If it looks longer than that, then an oncoming crew will be called in early to take the transport. We might call a crew in as much as three hours early or more, if the transport will be really long and run really late into the next day. Many times these are planned transports that dispatch knows about the night before; so they can often call the next-day crew to give ample warning that they'll have to come in early.
Mordechai Y. Scher
NRP, FP-C, RN
It's all about kind, competent patient care; and getting home safely to tell about it.