Lace has been noted in Mexico as early as the 1820's. It appears very close to the laces of Spain and the islands nearby such as Tenerife.  Many Mexican women had lace Mantillas worn over their heads, or when wealthy and could afford larger pieces of lace, over head and shoulders. And lace trim was almost a necessity on petticoats and gowns of women and children.
Drawn work with some similarities to Dresden work were also made in Mexico at the end of the 19th  century but it's labor intensive work led to it's decline by the early 20th century.
Large quantities of imported lace was also brought into Mexico and South America  for there was a voracious appetite for lace by both women and men.  But by the end of the 19th century so much lace was made in country that imports declined.
It is noted that lacemaking was a common training found in orphanages to sell and save money for the girls once they were discharged and began lives of their own.

photo of Mexican Lace from the Columbian Exposition of 1893