Mercenary Units.

[Yggdrasill]

Stradioti fought for other Italian city states, for France etc.
If you had the cash, they wouldn't ask you were you came from. Venice was their primary employer because of their extensive overseas possessions, they had an easy access to the Stradiot's primary recruiting grounds.

The Swiss and the HRE kind of hated each other. They were mercenaries and they would, in theory, fight for anybody, but they mainly fought for France which was their primary employer. They were found in other armies as well (Papal and other Italian statelets) but generally not in the armies of the HRE and the various German princes and bishops. If some did find themselves in there, they were usually the dregs of the Swiss war machine, and very inefficient, not worthy of their counterparts fighting for other nations.

The hatred began during the BAvarian wars, but they especially hated the HAbsburgs that had tried to conquer the Fedration at one point. Swiss confederation was very liberal and democratic republic and the generally disliked the feudal overlords.