In this paper, the author studies topologically transitive and hypercyclic composition operators on \(C(X,\mathbb{E})\), the space of \(\mathbb{E}\)-valued continuous functions on a separable locally compact metric space \((X,d)\), where \((\mathbb{E}, \| \cdot\| )\) is a finite dimensional normed vector space. In the main result, the author shows that, if \(G\) is a semigroup of continuous selfmaps of \(X\) such that every element of \(G\) is injective and the action of \(G\) on \(X\) is run-away, that is, for every compact subset \(K\) of \(X\), there exists an element \(f\) of \(G\) such that \(f(K)\cap K=\emptyset\), then the semigroup of composition operators induced by the elements of \(G\) is topologically transitive, hence it is hypercyclic.