Maybe? For what reason?
I get where you're coming from - but the whole notion that the "creator should get a share" comes only because of our current understanding of IP ownership. It hasn't always been this way, and odds are it won't always be this way. It may be the norm now, but is there an objectively good reason why it ought to remain the norm?
If I wrote a smash hit song when I was 20, should I get "a share" every time someone plays it 30 years from now? 50? Should my children get a share? There certainly isn't any notion of morality, in our society and most others, that this should be the case.
Which leaves copyright as economic policy. At that point the notion of whether or not we "should" do something is entirely based on the economic benefits it generates for society. Would I create more work for the public to enjoy if the copyright term was 10 years? 20? 30?
To take this example in a different direction - should the artist of a painting get a cut when you sell the painting to someone else? They did create it after all. People are still deriving value off of it, and even willing to pay for it, so why should the creator not get a share? The notion of whether or not someone should get a cut of anything is incredibly subjective, and is more related to the mores and standards of the time rather than any objective (or even subjective) notions of fairness/justice.