Correct call for this situation?

No reason the kill the play. Though the base is disloged during the remainder of the play everyone should assume the base is in it's correct position and play on.  Umpire will have to make a judgement calls on any plays at 3rd or touches. So if he is standing on the location of where 3rd base would normally be he should NOT be called out.

What if he's standing on the base when it's 2 or 3 feet away from where it should be?  Would you provide him protection if he clearly beat the throw / tag and is not trying to advance or go back?

I could see this only happening in youth league but never know when a base will pop off in high school or above.

If he's on the base and it's a few feet from where it should be, he's still safe.

Is that a rule or a judgement call? 

I doubt it matters.  It's not like this will ever happen again.

Rules, definitely.

For being in the same spot the base should be: 8-4-2h-1:

"if any base comes loose from its fastening when any runner contacts it, such runner cannot be tagged out because the base slides away from him."

Because he's in the position the base should be, you can't call him out.

For being on the base when it's dislodged:

8-4-2h: "Any runner is out when he is touched by a live ball securely held by a fielder or is touched by a fielder’s glove or hand with the live ball held therein, while the runner is not touching his base."

If he's out when he's not touching his base, then logic says he can't be out when he *is* touching it.

There aren't any case plays I can find that can back up my thoughts. Ultimately, it's going to fall on umpire discretion here, and I would have a hard time calling out a runner who is either touching the physical base or the place the base should be when it becomes dislodged. I also wouldn't uphold an appeal of missing a base if a runner does either of the two.

I enjoy this site for questions just like this.  I have learned so much about the rules I thought I knew all these years, friends call me to ask me if the call was right.  I'm not an umpire, and I don't play one on TV either, but you guys continue to educate me.

I have a recollection of a previous discussion regarding the bag coming loose.  It seems to me there was a point made about the base doesn't move, even if the bag representing it does.  So, I'm thinking 8-4-2h-1 referenced above, deals with the runner who slides in and causes the bag to come loose.  He is still on the "metaphysical, but actual" base, even though not in contact with the bag.  So, in the OP, runner from 1st is actually on the base, though not touching the bag, and should be safe.  However, if it were me, I'm like you guys and protecting the runner in either case, if he has made the spot, whether where it should be or where the bag is, before the play is made on him and tag applied.

Just my humble opinion from the cheap seats.

I should qualify my statements with the understanding that the base is only a few feet from its intended location. If it were more than that, then the "base" becomes the ground where the base should be.

Just like it was when we played with our friends in a large field somewhere without parents, coaches, and umpires

I have been learning the ropes doing youth leagues, and this situation is standard discussion during ground rules. Most youth leages dont have any base anchoring system.  Bottom line, if the base dislodges, the players needs to play to the original location.  There is judgement used... especially with young kids.  But on a close play, if the fielder is at the original location and the runner slides 3' up the line or into foul territory to the actual base, the runner will be called out.  Similar situation where the orange safety base is in foul territory at first.   If the fielder plays the orange bag, the runner is safe. 

When I was umpiring and training youth umpires I was instructed to teach that you play the location of where the base is supposed to be. I can't quote rules but I know that you play where the base should be for a fact.