What are the rules regarding baggage inspection during layovers?

Contrary to popular belief, “transit” doesn’t mean that you aren’t subject to the laws of the transiting country. So if what you are carrying is illegal — or simply of a nature that you might be held for secondary questioning, it isn’t worth it.

There have even been instances of people being arrested when their plane had to make a forced landing in a third country and they had prohibited items.

The safest: Instead, you might consider mailing yourself from the UK to Japan the prescription medicine you need and declaring it on your Japanese customs form at the airport as unaccompanied baggage (items sent separately). See link for how to do this.

Or buy a direct ticket.

If your layover is an airside transit, you probably won't pass through Russian immigration, won't see your checked luggage, and your checked luggage won't be examined by Russian customs officers. Whether yours is an airside transit will be determined by your flight schedule, airport terminals or transfers, your nationality, and Russian rules.

If, on the other hand, yours is not an airside transit, then you'll pass Russian Immigration and Customs, and your baggage (and your carryons, and you) may be searched.

In either case, and even if you have an airside transit, it's possible that you, or your checked or carryon baggage, may be searched anyway. This seems unlikely, but you, or your appearance or behavior, or travel history, or back-channel information available to the Russian authorities could lead to this result.

What they'd do after finding such meds is unknown. It probably runs the gamut from nothing, to arrest and criminal prosecution.

If you're going to be carrying such drugs, I think having only the amount you yourself will use, and having copies of the physician's orders — that is, the prescription(s) —- would be wise preparations.