The only really long lasting an workmanlike solution is to remove the trim, plane the back if necessary so it lays flat (Lots of this I see has cupped and pulled away) and renail with a little glue to fix it to the head jamb. I hate the idea of any type of caulk on varnished woodwork. If you don't get the joint tight, all kinds of dust and grit filters in through that seam, esp. in an old house.
I try to plan on taking these pieces off, adjust the miters (mine have mitered corners that became askew) and scrape the paint and caulk from that area, since it is a terrible thing to strip with scrapers, chemicals- you can never get it all.
Casey

Argh, Casey, thank you, although I was rather afraid of that answer. Removing and replaning certainly makes sense, even though the doorjambs, for instance, have skewed laterally enough to make a re-match well-nigh impossible without adding more wood to reach the header, or cutting the header down to where it's visibly shorter than its colleagues.

I will muddle along and fend off fillers of all kinds. Many kind thanks!