  We consider the spatial-temporal behavior of the Navier-Stokes flow past a
rigid body in $\mathbb{R}^3$. The present paper develops analysis in Lebesgue
spaces with anisotropic weights $(1+|x|)^\alpha(1+|x|-x_1)^\beta$, which
naturally arise in the asymptotic structure of fluid when the translational
velocity of the body is parallel to the $x_1$-direction. We derive
anisotropically weighted $L^q$-$L^r$ estimates for the Oseen semigroup in
exterior domains. As applications of those estimates, we study the
stability/attainability of the Navier-Stokes flow in anisotropically weighted
$L^q$ spaces to get the spatial-temporal behavior of nonstationary solutions.
