The hubs and larger stations have a dedicated group of agents who do nothing but retag bags all day. If you get rebooked they will try to get the bag to follow you but it depends on a number of things. That close to departure it's very unlikely that your bag will follow and almost guaranteed not to if it's already loaded on a flight. Generally once a bag is loaded it will take a fairly significant delay for bags to be offloaded. Smaller stations are a bit better about getting bags pulled since they don't have 300 flights to worry about. On rolling delays it's a bit of a pain because the agents are stuck trying to figure out if the flight will actually go out at the posted time. Sometimes people will get rebooked on flights leaving 2 hours after the posted time so the bags will be left on the original flight. If it keeps rolling past the new flight then it sucks but it also prevents people from rebooking, getting their bags moved to a new flight, and then them jumping back to the original flight 15 minutes before departure leaving no time to get the bags moved back.

When you get rebooked the agents are supposed to send a recheck request to get your bag retagged. Then those who are responsible for rerouting bags will see the request in the reroute program and retag accordingly. This is obviously a simplified way of explaining it but the short answer is yes if you change flights your bag should follow you. But again, depends on the situation on how likely it is to happen. During weather events where everything is delayed they just try to get bags moving as quickly as they can but they obviously won't be able to get to every request since they're dealing with so many bags.