  We investigate off-diagonal decay properties of the generalized Stokes
semigroup with bounded measurable coefficients on $\mathrm{L}^2_{\sigma}
(\mathbb{R}^d)$. Such estimates are well-known for elliptic equations in the
form of pointwise heat kernel bounds and for elliptic systems in the form of
integrated off-diagonal estimates. On our way to unveil this off-diagonal
behavior we prove resolvent estimates in Morrey spaces $\mathrm{L}^{2 , \nu}
(\mathbb{R}^d)$ with $0 \leq \nu < 2$.
