In my mind there is a difference between trying to isolate a player before their all-in is called by additional players and trying to run players out of the pot once a player has gone all-in and all players have called. For example, if a there were to be a rasie and an all-in 3-bet in front of me, and I felt that I held the best hand, I might re-shove to discourage the original raiser or any additional players from calling, thus isolating the all-in individual. Even if someone were to shove on the flop, with no one having gone all-in prior, I would probably re-shove to isolate them if I felt that I had the best hand.
Once an all-in has been called by multiple players, many players believe that there is an implied understanding that by checking the hand down, they increase the chances of eliminating the all-in player. This is often referred to as tournament strategy. While some players employ this thinkig from the beginning of a tournament, others only start to utilize this strategy as the bubble approaches or once the money has been reached.
Where some players feel that it is unnecessary to risk additional chips by betting into a dry side pot, almost all agree that you should never bluff at a dry side pot. Then there are some players that feel it's okay to do whatever it takes to try to take down each and every pot. The flaw in this thinking, in my opinion, is that it often works in favor of the all-in player. Frequently the player holding the best hand ends up folding it, and the all-in player ends up winning and having their tournament life spared. Of course sometimes, the player who reshoves their stack does end up having the best hand. But when the other player folds, they don't win a single chip more than they would have had they not risked the remainder of their stack. Then there is the possibility of shoving into a player that might have flopped an absolute monster hand, but be willing to check it down based on the principal that if you're a "check it down" player, then you always check it down the same way, whether there is a slight possibility that you have the best hand or no doubt that you have the absolute immortal nutz. In the last secenario, you would end up losing your entire stack re-shoving into a dry side pot when you could have seen the rest of the hand play out without risking any more chips than you already had by calling the first all-in.
I've had hands where I've called an all-in with A-Q and a second player called with A-8. I flopped Qs and checked down to the river where an A fell, giving the second caller top pair. Of course this gave me top two pair. I've played A-K and had the same card that the other guy a straight draw on the turn give me the nut straight. When a K hit on the river, they shoved their K high straight right into my broadway straight. Surprise! I've been holding the best hand. All you had to was check it down.
But I've also had people re-shove on me when they river two small pair on me. When I lose, my anger isn't because I lost the hand. It's because I could have bet it when I flopped the best hand, but instead decided that the priority was making sure the all-in player was eliminated, rather than starting a pissing war over a dry side pot. To be honest, I've started to get away from checking it down when a player is all-in lately, because of plays like this. I simply don't trust the other guy to continue to check it down if they should hit a better hand than me somewhere along the way. I've had people tell me, that I was supposed to check it down. I know. I would if I could have faith in other players to do the same.