The NIH allows non-US Universities and other institutions to apply for NIH grants. I don’t pay much attention to this issue so I don’t know how many.
What I am curious about is whether the PIs who review have noticed anything about these applications. How are they received in study sections you have attended? Is there a high bar for the unique environment or capabilities? 
I believe I was on study section during a transition where the foreign applications were essentially treated like domestic apps to where there was intense skepticism. This was around the mid-naughties, approximately a decade ago. 
I’m curious what you folks are seeing.