That would depend on the claim and whether it was deemed realistic they should conquer their throne. On average one would have to conclude that they'd live at least as comfortable as local nobility would.

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We have a few examples in history:

For example Edward IV spent some time in Flanders during the war of the roses. He lived in every comfort and was supported by his brother-in-law the duke of Burgundy.

The old pretender and his son lived quite comfortably in France mainly, but one has to realize that once their chance of regaining their throne diminished, so did their housing situation. Henry Stuart was the last real pretender and he joined the church.

Louis XVIII is a weird example. He was shoved through Europe and welcomed no where. Very strange, but according to some of his fellow monarchs it was the fault of his family they all had to deal with the French Revolution and therefore they didn't want anything to do with him.

Napoleon III was on very good terms with Queen Victoria so when he had to step down he led a very comfortable life in England. There were even talks of his son marrying princess Beatrice.

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So coming back to the start of our story, it depends on a lot of factors, but mainly two:

1. The chance one has to regain/conquer the throne
2. The relationship with the monarch they're "staying" with