The modularity of termination and confluence properties of term rewriting systems has been extensively studied, for disjoint unions and other more types of combinations. However, for rewriting under strategies the theory is less well explored. Here we extend the modularity analysis of termination properties systematically to (variants of) innermost and outermost rewriting. It turns out — as expected — that in essence innermost rewriting behaves nicely w.r.t. modularity of termination properties, whereas this is not at all the case for outermost rewriting, at least not without further assumptions.