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Let X and Y be metric spaces and E, F be Banach spaces. Suppose that both X and Y are realcompact, or both E, F are realcompact. The zero set of a vector-valued function f is denoted by z\left( f \right). A linear bijection T between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions if
respectively. Every zero-set containment preserver, and every nonvanishing function preserver when \dim\,E\,=\,\dim\,F\,<\,+\infty, is a weighted composition operator \left( Tf \right)\left( y \right)\,=\,{{J}_{y}}\left( f\left( \tau \left( y \right) \right) \right). We show that the map \tau \,:\,Y\,\to \,X is a locally (little) Lipschitz homeomorphism.
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