Took an untraded villager and had them in a minecart next to a zombie (nametagged) in a minecart.  The villager was converted to a zombie and left minecart.  Tried to get villager into cart, and was unsuccessful several times.  Ended up putting rails with minecarts in a small room.  Tried for a 6th time, converted, villager could push carts around.  Not connecting and locking into minecarts.

A "wild" or found zombie villager could be captured with a cart.  Weakened, cured, and would remain in cart.