The seriously poor also need cash for rent and to find a new place, and a new job. Those don't materialize out of nowhere. Having no savings is still a significant barrier.2) The financial cost of moving is proportional to how much stuff you have. The seriously poor can move for the cost of gasoline or a bus ticket.
Sure, if you only have a knapsack and live in the streets you can move for the cost of a bus ticket. Otherwise you need money. Maybe not $25k, but not nothing either. Most people don't have cash to cover a $1k emergency, and I doubt anyone that owns a bed or a desk, and rents a place, can move for less than that.