The aliens don't have to do a thing. If the Earth's industrial civilization dies, then the outposts on the Moon are essentially helpless. They are cut off from the tools and spare parts they need in order for their systems to continue working and the gradual cascade of system failures will kill them in an matter of years, perhaps a few decades at most.
The people trapped on the outposts might stretch out their existence by trying to bootstrap their own industrial civilization, but lacking the mass of tools and infrastructure surrounding the real thing, they will be able to mostly cobble together simple things like plumbing and basic electrical infrastructure. Electronics, and especially things like modern integrated circuitry will be impossible to replicate, and as circuit boards and microprocessors fail, the devices dependent on them will also fail.
This will be especially tragic since much of the accumulated knowledge that the people on the Moon will be trying to draw on will be stored in electronic format, so accessing help files or watching YouTube videos on how to fix systems will become more and more difficult as time passes. The other issue is there will be a small pool of people, so any accident that kills a person will also remove their accumulated skills and experience from the available pool, making the remainder less robust and able to deal with emergency situations. (As an aside, this is thought to be one of the reasons the Neanderthals died out; their bands were much smaller than those of our Ancestors, so the death of one man or woman would leave them with large skill gaps and make it harder to survive in late Ice Age Europe).
The people on the Moon were obviously selected for their skills, smarts and ability to work under difficult conditions, so they will likely go down fighting their failing systems to the bitter end, or try to return to Earth if they believe the pathogen or whatever killed the Earth has passed as well. You have the basis for a pretty good story right there.