I think what the OP is asking isn't about when to take medicine in the new place, but how to do the transition.
For example, Hong Kong is 12 hours ahead of New York. To keep taking a particular medicine in the morning, say 7am, one can do either of two things ONCE: stretch to 36 hours between taking it, or cut it to 12 hours. Same thing coming back.
Or do the adjustment over 2-3 days. So, instead of doing 36 hours, then 24 hours; one may time it to take after 30 hours, then another 30 hours. Or instead of 12, then 24; one can do 18 and 18.
It's an interesting question. One may want to ask his/her doctor or pharmacist about it. It will depend on the medicine as well.