  For $T$ a compact torus and $E_T^*$ a generalized $T$-equivariant cohomology
theory, we provide a systematic framework for computing $E_T^*$ in the context
of equivariantly stratified smooth complex projective varieties. This allows us
to explicitly compute $E_T^*(X)$ as an $E_T^*(\text{pt})$-module when $X$ is a
direct limit of smooth complex projective $T_{\mathbb{C}}$-varieties with
finitely many $T$-fixed points and $E_T^*$ is one of $H_T^*(\cdot;\mathbb{Z})$,
$K_T^*$, and $MU_T^*$. We perform this computation on the affine Grassmannian
of a complex semisimple group.
