Well there's another intriguing question Pallas!!! You do have a way of posing some pretty compelling questions (I'm still completely enthralled in the Nauvoo Legion one!! BTW did any of that information prove helpful?).
Actually, the Landsknechts were an interesting military phenomenon that corresponded with mercenary forces who were employed by various countries and principalities in times of war. There is a good deal of information on them here in a number of discussions on the weapons used by these Landsknechts.
Since these troops for hire were used for war, they would not have engaged with these expeditions intended to expand the empire of Spain, and ostensibly a religious mission as well. While presumably carried out in somewhat the same sense, the expeditions of France and England were also just that, and not war.
Regarding your interest in the Landsknechts, what has your research revealed so far on them? Are you interested in them specifically, or in these expeditions of discovery?
Thanks for keeping things interesting!!!
All the best,
Jim