A geometric (n₄) configuration is a collection of n points and n lines, usually in the Euclidean plane, so that every point lies on four lines and every line passes through four points. This paper introduces a new class of movable ((5m)₄) configurations---that is, configurations which admit a continuous family of realizations fixing four points in general position but moving at least one other point---including the smallest known movable (n₄) configuration.