Learn English – “I’ll stake you. Pay me back next time.”

The more common option is indeed let's go, but let's move works as well - with a slightly different meaning!

Let's move can mean several things:

• Let's displace objects (or people) from one place to another. (The soldiers)
• Let's go and live somewhere else. (We're moving next month)
• Let's get a move on - let's physically move our bodies. (because we want to go somewhere)

However, for the last meaning, I believe the more idiomatic let's get a move on or let's get moving are more commonly used.

It is not just used when you want to go somewhere, you can also say:

Let's move! The house won't clean itself.

But more idiomatically:

Let's get a move on! The house won't clean itself.
Let's get moving! The house won't clean itself.

As noted by commenters, this may depend on the culture; but my own opinion is that it can cover even fairly distant relations, provided that the family relationship is the reason for the continuity. So, if operation is transferred from father to son-in-law, and then to son-in-law's younger brother, then to that younger brother's granddaughter, and then to that granddaughter's cousin, and so on, then it's still been "run by the same family", even though at this point the original manager and the current manager are essentially unrelated. On the other hand, if the enterprise is put up for auction, and the second cousin of the current owner-and-operator happens to place the winning bid, I'm not so sure that I would count that. (But "the same family" is sufficiently vague that the new owner could probably get away with making the claim.)