James Keith would be a historical example of an officer that served in multiple armies. In the 1600s, Norwegian seaman Kurt Sivertsen served in the Dutch, Venetian and Dano-Norwegian navies under the name Cort Adeler. Carl von Clausewitz served in the Russian army for a while during the Napoleonic Wars. And even in the mid-19th century, it seems that navies would recruit sailors without regard to nationality, same as merchant vessels at the same time. These days, it is normally a requirement for military service that you are a citizen of whichever country whose military you are trying to join. Exceptions usually only apply to citizens of closely allied states.

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In short: When and why did citizenship become a standard requirement for military service?