Base change for complex of coherent sheaves

I wonder if the base change for a flat complex of coherent sheaves works. Namely, let $f : X \rightarrow Y$ be a proper morphism of varieties over $k$. Let $\mathcal{F}^{\bullet}$ be a bounded complex of coherent sheaves on $X$ such that all the cohomology sheaves $\mathcal{H}^{\bullet}(\mathcal{F}^{\bullet})$ are flat over $Y$. Assume that the dimension of $\mathcal{H}^i (\mathrm{R} \Gamma(\mathcal{F}^{\bullet} \otimes k(y))$ is constant when $y$ varies in $Y$. Do we have a base-change formula:

$$\mathcal{H}^i (\mathrm{R} \Gamma(\mathcal{F}^{\bullet} \otimes k(y)) = \mathcal{H}^i (\mathrm{R}f_* (\mathcal{F}^{\bullet})) \otimes k(y),$$ for all $y \in Y$ ?