I'm not sure where the figure of a 1,000 Mexican soldiers comes from. To the best of my knowledge, while the Mexican government certainly provided considerable military supplies to the Republic, they did not go so far as to send military personnel to Spain. Complicating this picture is the rather transparent Soviet attempts to disguise their own military presence in Spain by referring to their personel as Mexicans (the International Brigade volunteers called them 'mexicanskis'). So, it's entirely possible that various sources at the time and since have mistakenly concluded that Soviet soldiers (of which there were about 1,500) were in fact Mexicans.

Individual Mexicans certainly did volunteer to fight though. However, like many Latin American volunteers, they faced no language barriers in joining the Spanish Republican armed forces rather than specific international units - indeed, it would likely have been easier to join a homgenously Spanish-speaking unit. Moreover, as the Mexican government was quite willing to allow their citizens to go, unlike virtually every other government in the world, there was no need for secrecy. This all meant that the Communist Party played a relatively small role in recruiting volunteers compared to other contexts, and that relatively few of the Mexican volunteers joined the International Brigades (though some - especially Mexican-Americans - certainly did). This also means though, that we know relatively little about them, as they don't appear in some of the key source bases about the foreign volunteers in Spain. Estimates vary from about 150 to 800 such volunteers spread across much of the Republican armed forces. I'm not aware of any who joined the militias of either the POUM or anarchists - there may well have been some, of course, but these units are poorly reflected in the source base, and many Mexican names would not stand out in the records that do survive. Keep in mind that the militia phase of the war was quite short, and the militia system was being wound down by the time most volunteers actually made it to Spain (especially from as far away as Mexico).

All of this means that a satsifactory answer to your question is rather difficult to provide. Unlike those who served primarily in the International Brigades, it's very difficult to point to specific battles or fronts in which Mexicans played an obvious role, because they were all over the place. I'd suggest checking out the recent book by Mario Ojeda Revah ('Mexico and the Spanish Civil War'), which should be available in either English or Spanish. It concentrates more on the Mexican side of things, but has a chapter on Mexicans in Spain, and gives some details about individual volunteers.