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Can someone give me an example of a closed smooth oriented manifold $M$ and an orientation preserving diffeomorphism $f:M\rightarrow M$ such that $f^k$ is isotopic to the identity for some $k\geq 1$, but $f$ is not isotopic to any finite order diffeomorphism? Similarly for homeomorphisms, and also for homotopies.

Of course, Kerckhoff's famous theorem says that this is impossible in dimension $2$.

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This is discussed at great length (with copious references) in Jonathan Block's and Shmuel Weinberger's paper. The title of the paper is the suggestive "On the generalized Nielsen realization problem" (comm math helv, 2008)

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