Nehmo wrote:

I have an ornate wood interior staircase that is several decades old.
Many of the railing joints are loose to one degree or another, and I'm
going to make them more solid. It would be chore to disassemble &
reassemble the entire railing assembly to repair, so I'm trying to come
up with another method.
I've never done this, but is it a reasonable idea to drill holes, say
3/16" diameter, to every joint (in places that aren't very visible) ,
and then use a syringe to pump some polyurethane glue into the joints?
In some places I could also drill holes and insert trim screws.

Recommendations welcome.

Depends on what the cause of the looseness was originally and how bad
the damage is.

I'd be reluctant to do something that drastic as my first step primarily
because if it doesn't work well anything from then on will be nearly
impossible w/o destroying it. The first rule in restoration as in
doctoring is "do no harm"...