Let $\mathbfΓ = (V,E)$ be a (non-trivial) finite graph with $λ: E \rightarrow \mathbb{R}_{+}$, an edge labelling of $\mathbfΓ$. Let $ρ: V\rightarrow \mathbb{R}^{2}$ be a map which preserves the edge labelling. The graph $\mathbfΓ$ is said to be flexible if there exists an infinite number of such maps (upto equivalence by rigid transformations) and it is said to be movable if there exists an infinite number of injective maps. We study movability of Cayley graphs and construct regular moving graphs of all degrees. Further, we give explicit constructions of "dense", movable graphs.