A graph $\Gamma$ is $k$-CS-transitive, for a positive integer $k$, if for any two connected isomorphic induced subgraphs $A$ and $B$ of $\Gamma$, each of size $k$, there is an automorphism of $\Gamma$ taking $A$ to $B$. The graph is called $k$-CS-homogeneous if any isomorphism between two connected induced subgraphs of size $k$ extends to an automorphism. We consider locally-finite infinite $k$-CS-homogeneous and $k$-CS-transitive graphs. We classify those that are $3$-CS-transitive (resp. homogeneous) and have more than one end. In particular, the $3$-CS-homogeneous graphs with more than one end are precisely Macpherson's locally finite distance transitive graphs. The $3$-CS-transitive but non-homogeneous graphs come in two classes. The first are line graphs of semiregular trees with valencies $2$ and $m$ (where $m$ is a positive integer), while the second is a class of graphs built up from copies of the complete graph $K_4$, which we describe.