If the change is only a few minutes they likely won't do anything. I'm not sure what "significant" means but almost certainly a "few" minutes wouldn't qualify. The definition of significant to me would depend somewhat on the schedule and the number of flights per day.
I another forum a user was complaining that Southwest changed his flight to 10 minutes later and that somehow messed up his entire schedule. Southwest wouldn't make any accommodation, including changing him to another flight. In that case a flight arriving 10 minutes late wouldn't even be considered late. The consensus of opinion there was that it wasn't a significant change and Southwest traditionally has several flights a day on it's routes