Does migration exacerbate (albeit delay) the problem of aging, economically-inactive populations in Western societies? As a layman, I've always understood that there is a general consensus amongst economists that "immigration is economically beneficial" to rich Western countries (like Britain) who have aging populations because the influx of young economically active workers contributes to the economy (and tax coffers) and thereby supports the old economically inactive population.

But this seems quite obviously short-sighted.  Those immigrants will themselves grow old—and assuming that they remain in their new host countries, they will add to the economic burden of the elderly populations for future generations.

Is such immigration therefore not just kicking today's problems into the future, where it will have compounded and be exponentially worse?
I don’t really see the connection you are drawing. As @Fizz has pointed out, immigration usually happens over time and maybe when immigrants grow old there will be more immigrants to support all of the new old. Likewise, in many countries immigrants tend to have more children than non-immigrants, so that will attenuate the aging problem. 

If anything it should be something to check case-by-case, but I don’t see any causal relationship between immigration and the “aging problem”. I see how just mechanically these two are related, but I don’t expect the relationship to have any particular systematic pattern. 

It feels to me like you are saying that taking more exams is bound to lower your average grade. Of course more exams will impact your average grade, but it really can go up or down depending on many other factors.