Yes, all training towards any FAA license or a rating can be done anywhere in the world, in any aircraft, with any appropriately authorised instructor.
Except that you need to be signed off by an FAA CFI/CFII as ready for the checkride, plus you need to have flown 3hrs with him within the preceeding 60 days. Those bits need to be done with a pukka FAA instructor.
All other flights e.g. the various cross-countries, can be done in Mongolia
On occassions, questions
might be asked about some long ago logged flights, so having a neatly written up logbook, annotated by the instructor for any instructional flights, is a good idea. Before I did my CPL checkride, the DPE (a bloke from the USA) spent ages going through my logbook in minute detail, checking I had all the qualifying flights, and the FAA instructor did the same a few hours before.