Should command teams be replaced as a whole or have overlapping time?

As SGM Erik Marquez states, there are advantages to either way. However, what you see most often is the staggered approach. And I think that is the way it should be.

The first reason is continuity. The Outgoing CSM briefs the incoming CDR and gets him up to speed over several months. The CDR (no longer the NEW CDR) briefs the incoming CSM and gets her up to speed over several months. This local & unit knowledge is key to making effective decisions from the start. It also avoids the unit thinking "great, they're doing the new sheriff routine."

The second reason is focus. This is more so on the AC than RC side, but going to a new unit involves 1,001 things to take care of on a personal/family level - especially if you PCSed. Overlap allows for all of those things that theoretically happen on permissive TDY - but don't because you want to get to your command.... Also, there are all the evals at the old unit to close out, etc.

When it would make sense to replace both simultaneously is when the higher command WANTS a new sheriff in town. There could be a few valid reasons for this including ineffective command team, toxic (that's different!!) command team, a severe personality mismatch between the command team and its next higher level. There are others, but those would be the big ones. None of these would be a "on schedule" transition. (They might not be severe enough to formally result in a relief, they might just play with dates for various reasons.)

I think that they should over lap a little. Gives the in coming commander time to the swing of the ropes and the happenings of the unit. The way things run and time to see what needs changed and implement the change in an orderly fashion. Not just all of a sudden. I'm the new sheriff in town this is law and this is the way it WILL be done. 

In most cases Id prefer to see some overlap at the company level, and team swap at the BN and higher.
In some cases, circumstance leaves no choice, people come and go for a myriad of reasons, many of them not well timed and not planned.

I've seen it both ways as well as many ways in between.

There are advantages in the all at once, and advantages for the staggered.

It really is best left to the senior commander either outgoing or newly assigned (who then observed for a period of time as available)That commander should have the benefit of observing the unit, it's current leaders and the leaders he intends on placing in those position to decide what dynamic will work best.