Here is one example.
Let $M^3$ be a hyperbolic manifold. Consider the moduli space curvature $-1$ metrics on $M^3$ modulo $Diff_0(M^3)$. This is a point. Conclusion: every diffeomorphism of $M^3\to M^3$ is homotopic to an isometry.
Here is one example.
Let $M^3$ be a hyperbolic manifold. Consider the moduli space curvature $-1$ metrics on $M^3$ modulo $Diff_0(M^3)$. This is a point. Conclusion: every diffeomorphism of $M^3\to M^3$ is homotopic to an isometry.