Ouch, this is quite annoying, I just hit this bug.

Are there any workarounds? Or should I go into the imported scene and shift it in time to the same position where it's being used in the master scene?

Sergey, I'd be more than willing to write a patch, if you let me know the desired behaviour. About causing regressions, don't blend file contain information on what blender version they were created in? If we had that, then we could correct the behaviour for new projects but keep the old behaviour when loading old projects. Or am I missing some complication?

Sergey, forget my last question; I see that blender files are supposed to be both backward and forward compatible, so we cannot play tricks based on the blender version.

What if we add another checkbox just below the text that now says "Original frame range: <A-B>" and call it something like "Shift all timings to new range". Well, I'm sure better names can be found, but this is not too bad, IMHO.

I've spent some time trying to undestand how this works, and I got something which is more or less working, though for some reason the image in the sequencer sometimes is updated in the wrong way.

Can some blender developer comment on what I'm doing (patch attached)?

I suspect that my approach will fail miserably if the same scene in used twice (at different start times) in another scene, because AFAIU the animations are evaluated only once. What would the correct approach be?