If the crossing's short enough, the shoe on the rear truck keeps contact, the shoe on the front drop off, then picks up, then the rear comes off, and picks up again. Thus, there's always a shoe on the rail. I think married pairs will share power between each other still. There's no trainlined power bus like the MP-54s had (yes, they did at one time).
If the train's going fast enough, it doesn't matter, unless the first car's batteries are low, in which case the brakes can dump. I've been on amazingly dead cars that didn't, though. One time during the 'summer of hell' the other year, the first car had no MA, no A/C, no lights, sluggish doors, yet still the train kept going. I think about the only thing the car WAS doing right was taking power - you could hear the controller klunking away under the car...
If the crossing's long enough, yes, the car loses power over the crossing. Jamacia's known for this, esp if the train stops and the MA set winds down - everyone gets to sit in the dark until the car hits the third rail again and the MA spins up and all.