GG5: 音楽は最も国際的な芸術の 1 つである. Music is one of the most cosmopolitan of the arts.
KOD: 国際的潮流 an international [a worldwide] trend.

Of the 22 GG5/KOD snippets 20 use "international" somewhere. I think it helps to cast the net widely here. Probably "cross-border" is OK in some contexts, e.g. travel.

@@ -18,0 +19,2 @@
+<gloss>worldwide</gloss>
+<gloss>cross-border</gloss>

Re: 国際的な[人の往来], ah, yeah, that makes more 
sense. But doesn't cross-border work quite 
well with that parsing in this sentence? 
"cross-border comings and goings of people" 
(if you excuse the literalness) is slightly 
more on point than "international comings and 
goings of people", no?

if there's several glosses showing how widely 
the word can be used, I don't mind 
"cosmopolitan", but if it's just 
"international" and "cosmpolitan", should 
there maybe be 2 senses? I feel the meaning 
could easily be misunderstood as it is.

I agree it doesn't really work that well. That MOFA translation is rather clunky. GT's version is rather better,

@@ -19 +18,0 @@
-<gloss>cross-border</gloss>

In that sentence that Marcus quoted, isn't 国際的 modifying 人の往来? I'd parse it as 国際的な[人の往来], not [国際的な人]の往来. I don't think "cross-border" is needed.
I agree with Frazer that "universal" looks out of place here.

@@ -20 +19,0 @@
-<gloss>universal</gloss>

Even if it is in GG5, I really don’t think the “universal” gloss belongs here.