We study mixing properties (topological mixing and weak mixing of arbitrary order) for nonautonomous linear dynamical systems that are induced by the corresponding dynamics on certain invariant sets. The kinds of nonautonomous systems considered here can be defined using a sequence $(T_i)_{i\in\mathbb{N}}$ of linear operators $T_i:X \to X$ on a topological vector space $X$ such that there is an invariant set $Y$ for which the dynamics restricted to $Y$ satisfies a certain mixing property. We then obtain the corresponding mixing property on the closed linear span of $Y$. We also prove that the class of nonautonomous linear dynamical systems that are weakly mixing of order $n$ contains strictly the corresponding class with the weak mixing property of order $n+1$.