Mixing Fantastic Creatures from different Cultures

Fantasy RPG settings have almost always drawn inspirations from real world cultures, including real world folklore and mythology. This also includes fantastic creatures, which are a staple of monster manualy since the earliest days of D&D.

And in recent years, there have been more and more efforts to draw TTRPG inspiration from non-European sources - a trend I approve of, since we can always use more diverse sources of inspiration.

However, I am curious how you all use creatures from different cultures in your fantasy campaigns and worldbuilding. I could see a few different approaches:

Do you assume "counterpart fantasy cultures" for your world? I.e. is there a Not!Europe, Not!Asia, Not!Africa, Not!Mesoamerica etc., each of which has "culturally appropriate" supernatural creatures? In such a situation, supernatural creatures would largely stick to their "home region", and occasional "guest stars" in other regions would be noteworthy. This works well, but it does mean you are usually limiting yourself to a "culturally appropriate" subset of the creatures in the books.

Alternately, you could try to integrate these creatures into whatever cultural region your campaign focuses on. In my view, this would require some rewriting in how these creatures are portrayed and/or named - if your region has a culturally English or Germanic slant, then having creatures that are called "Sramana", "Betobeto-san", or "Tzitzimitl" (to pick three examples from the recent Monster Core 2 for Pathfinder) will seem strange.

I feel there is no "right" or "wrong" approach to this, but I am curious how you are tackling this issue.