Can you be cleared to cross a runway *behind* an aircraft on takeoff? Taking KEWR as an example, let us say that aircraft are taking off from 4L at N and landing on 4R.  Can I be cleared to cross 4L at say V, behind the aircraft taking off on 4L, without interrupting the departure flow? Or will the runway need to be clear full-length to let me cross it?
You can cross behind the aircraft on departure roll. This type of operation is done heavily at ATL, on the middle complex during an east flow. Aircraft will depart  runway 9L at M2, and arrivals from either of the south runway will cross behind them the whole time at M1, to get up to the gates.