Feudal lords have obligations to any liege from whom they hold fiefs in fealty. If a lord holds fiefs in two kingdoms, he is under feudal obligations to both sovereigns. This applies even if a vassal inherits a kingdom and become a king in his own right. For example, the Kings of England remained nominal feudal vassals to the Crown of France for their continental possessions.
This is evidenced by such instances as when the Lusignans (who were vassals of King John in his capacity as Count of Poitou), feeling wronged by John, appealed directly to King Philip II in Paris. Moreover, when John refused a royal summon to the French court, Philip declared his French possessions forfeit and seized them. In the other cases, the kings of England paid homage (usually kicking and screaming all the way) to the kings of France over their continental possessions.
The most fitting example to your question, though, would be the Dukes of Burgundy. While nominally a French fief, the lords of Burgundy acquired many territories that were nominally part of the Holy Roman Empire. In AD 1416, John the Fearless paid homage to the Emperor for Franche Comté, which was an Imperial fief, while the French King Charles VII later exempted his son from paying homage in an agreement (i.e. normally the Duke of Burgundy was obliged to paid homage).
Thus we may observe that, his position in the realm of England notwithstanding, King John was legally subject to the feudal overlordship of Philip with regards to his holdings in France. Of course, note that in practice a vassal with many holdings might be too powerful for the nominal liege to do anything about.