Depending on the institution they work for, different rules may apply. But in general, scientists would be encouraged to seek funding from wherever it is available.

Probably way harder to convice govts to give their limited amount of grants to foreign universities instead of their own. It is not like this is the only important science to be funded. If you choose to fund this - then it means you need to not fund something else.

I think it's rare without strings.

There are some cooperation grants, where you must have researchers from two countries (where you are and where the money comes from) and perhaps part of the money must be used to pay travel and stays of some of the researches in the other country, in both directions.

It is certainly possible to try to get another government to fund it, but the NIH is the largest source of funding for health science research in the world, both for domestic U.S. research and also international research.

So is it theoretically possible? Yes. Is it likely, especially for expensive clinical trials? Not really.

I suppose there's always the private sector, but there's much more money in diet pills than children with lifelong illnesses.