Here's the thing— we're assuming advanced civilizations want to expand in the first place. Why?

Life in OUR planet was shaped by competitive evolution and we have reason to believe life in other planets has been as well— it's how natural selection works after all, and at least in our planet it's proven to be an effective way to go from primordial soup to complex, intelligent life that can out-smart its competition hoard all the resources.

But how do we know that this is not a behaviour that civilizations, well, outgrow once they've hoarded all resources?

Unbound growth is an effective strategy in a competitive scenario like the evolutive stage of species, in large part because it's also one of simple incentives. When a species becomes dominant though, it is very dumb, because the species will just end up reaching the limit of available resources and burning through them, thus killing themselves, like bacteria in a petri-dish multiplying until all the food has been eaten and they all starve to death.

The argument goes, that's when a civilization would expand. Those bacteria would try to find another petri-dish. The species finds more resources so it can keep growing, but, can this incentive of "we have to keep growing" stand the test of time?

Simply from a practical standpoint, Alpha Centauri is 104691 times further than Mars. The technological gap between sending a colony to a planet in your system, versus one in another star is gargantuan.

First of all let's consider that planets are all ballpark similar in size and resources, therefore value. Is the investment even worth it? Is traveling 104691 times more for the same resources a stronger incentive that simply learning to find equilibrium with the resources easily available in your system?

Even if for the sake of argument the civilization decides it is, could they even achieve it before running out their available resources? It'd be a race against the clock.

There is reason believe the civilization would be forced to find stability and equilibrium within their existing system before developing the technology to be able to feasibly colonize a neighboring star system for the purpose of extracting resources. And if not to extract resources, why put the effort to continue expanding?

Fear? Of what? A nonexistent galactic empire?

Pride? If their history is anything similar to ours (which, since we're talking about similar motivations, is perhaps a feasible assumption to make), they would've learned colonizing far and wide does anything but promote a stable, unified, timeless nation.

Therefore, my guess is one of two things might happen:

1. They achieve stability because it's the only way to survive, and the motivation to expand to another system dies off. Probe and explore, perhaps. But if in a stable system, there's no incentive to introduce instability once again by adding another habitat. On the contrary, the prospect of instability is incentive to not.

2. They don't achieve stability in their system, and so collapse.

In a nutshell, I don't think there's incentive for unbound expansion once stability has been achieved, and I don't think unbound expansion can be achieved before stability. Of course, this is all just my opinion, but it's worth thinking perhaps the entire premise of extrapolating primitive evolutive behaviors to an advanced civilization is out of frame.

Consider the future world of Brave New World by Huxley, or even the Time Machine by H. G. Wells. Neither seem to have galactic empires; instead opting for (very distinct) forms of stability (and decay).