how to move multiple layers on the timeline?

It is a bit unclear from the gif, but I think that the problem may be that you are trying to move the primary and connected clip, but not dropping that into the primary storyline. That way it does not work.

If you move them to another location in the primary, it is fine.

If these two stacked clips are to be moved together and placed above the primary, you can first turn them into one compound clip, and drag the compound. A compound clip is treated just like any other clip, and can be placed anywhere.

I don't think it is wrong. If clip B is connected to clip A, and you move A somewhere in the primary, clip B appropriately goes along, respecting the connection point. But if you try to move A somewhere else, effectively turning it into a connected clip, where should clip B connect to?

Note that this will effectively break the connection between B and A, because you can't connect to a connected clip, only to the primary.

I can think of two solutions: 1) Move A using tilde key, and then move B; 2) Make A and B a compound clip

Louis got the details correct.

I'm trying this in 10.8 and getting the same results as the OP.

Odd, I can't move a clip out of the primary timeline with connected clips, but I can move one with no connected clips, AND I can move it using the Tilde key to override the connections. This is really inconsistent behavior. Is this a bug or did Apple change this purposefully in 10.8?

When moving a main storyline clip along with connected clips, FCP won't let you.

Not sure you that's right Ben. Moving storyline clips with their connected clips has always been a feature of FCP. I'm probably misunderstanding.