The term Paddy Wagon actually goes back to the early 1700s in southern states when big tough unskilled Irish immigrants were hired to pursue runaway slaves or transport slaves to new owners or otherwise control defiant slaves. These men were called "Paddy Rollers". These Paddy Rollers were given horse drawn wagons to use to transport the slaves hence "Paddy Wagons".
The slaves were thrown into the Paddy Wagon by the Irish Paddy Rollers.