Moving averages on convex metric spaces

We show that the moving average induced by a contractive mean on a complete metric space converges. This general scheme of convergence covers the moving arithmetic average studied recently by Bauschke, Sarada and Wang (also by Borwein, Borwein and Sims). Significant portions of the derivation can be carried out in general convex metric spaces, which means that the results have broader applications beyond the setting of Banach spaces.