Corporate jet First Officer reported an altitude deviation on approach in a foreign location; which resulted in descending below a charted altitude in mountainous terrain. The flight climbed to the correct published altitude then continued the approach.

Narrative

We were assigned the RETOS 1A STAR into MMMY (MTY) and were on profile; on speed and in good shape. The controller took us off the route giving us shortcuts which we normally appreciate but in this case it added to workload. This is mountainous terrain and communication was more difficult than normal due to the language differences. The non-flying pilot was off frequency for a moment to get the ATIS when were were cleared direct to SIKRO with unrestricted descent to 14000. When the pilot returned to frequency; he questioned the altitude (as he should near terrain and when on an arrival with step down altitudes.) I suggested he confirm so he could hear them for himself and make sure we all agreed what the clearance was. Shortly thereafter we slowed down to 220 kts to be able to configure for landing as needed. We were then given another shortcut to MY601 and told to descent unrestricted again to 7000. Weather was not bad but we could not see the airport at this time. We began configuring with flaps and slowing further to 170 kts. Then the controller took us off the arrival again and gave us a heading to intercept the final inside of TASNA. The non-flying pilot set the altitude preselect for 3300 as that is a mandatory altitude and we were descending. I was trying to slow the aircraft and didn't catch that the airplane descended to 2900 feet. The non-flying pilot called the deviation and we climbed back up to 3700 which was the published altitude at MY603. We were below altitude for about 1 minute - we had a view of the ground and knew where we were so I was smooth about the correction. We crossed MY604 at 3300 as published. The pilot (me) should have caught the altitude but the root cause of this situation was being taken off an arrival procedure which caused confusion about what altitudes had to be met and which were unrestricted. The non-flying pilot was changing the altitude preselect and we were going back and forth between VS and VNAV on this arrival and approach. This was more confusing because we were in Mexico and language/accents were an issue. We were trying to be precise with our communications and being active listeners. Actually; whether this was in Mexico or the USA; my opinion is that a controller should either give a pilot radar vectors for the entire arrival OR leave a pilot on the published arrival/approach. Making changes near the airport adds workload; stress and leads to errors. Had we been left to cross TASNA at 4800 we would not have had this issue. I appreciate that the controller was trying to expedite traffic and get us on the ground sooner but in this case I would have rather flown the published procedure.