In this work, we present an approach for the efficient treatment of parametrized geometries in the context of proper orthogonal decomposition (POD)-Galerkin reduced order methods based on finite-volume full order approximations. On the contrary to what is normally done in the framework of finite-element reduced order methods, different geometries are not mapped to a common reference domain: the method relies on basis functions defined on an average deformed configuration and makes use of the discrete empirical interpolation method to handle together nonaffinity of the parametrization and nonlinearities. In the first numerical example, different mesh motion strategies, based on a Laplacian smoothing technique and on a radial basis function approach, are analyzed and compared on a heat transfer problem. Particular attention is devoted to the role of the nonorthogonal correction. In the second numerical example, the methodology is tested on a geometrically parametrized incompressible Navier-Stokes problem. In this case, the reduced order model is constructed following the same segregated approach used at the full order level.