We introduce node-variant graph filters, which allow the simultaneous implementation of multiple (regular) graph filters at different nodes, and study their design to implement arbitrary linear transformations between graph signals. Node-variant graph filters can be implemented distributedly, making them suitable for networked settings. We determine spectral conditions under which a specific linear transformation can be implemented perfectly and, for the cases where perfect implementation is infeasible, the design of optimal approximations for different error metrics is analyzed. We demonstrate the practical relevance of the developed framework by studying the application of node-variant graph filters for analog network coding.