Wrinkling instabilities of thin elastic sheets can be used to generate periodic structures over a wide range of length scales. Viscosity of the thin elastic sheet or its surrounding medium has been shown to be responsible for dynamic processes. We here consider wrinkling of fluid deformable surfaces. In contrast with thin elastic sheets, with in-plane and out-of-plane elasticity, these surfaces are characterized by in-plane viscous flow and out-of-plane elasticity and have been established as model systems for biomembranes and cellular sheets. We use this hydrodynamic theory and numerically explore the formation of wrinkles and their coarsening, either by a continuous reduction of the enclosed volume or by the continuous increase of the surface area. Both lead to almost identical results for wrinkle formation and the coarsening process, for which a scaling law for the wavenumber is obtained for a broad range of surface viscosity and rate of change of volume or area. However, for large Reynolds numbers and small changes in volume or area, wrinkling can be suppressed and surface hydrodynamics allows for global shape changes following the minimal energy configurations of the Helfrich energy for corresponding reduced volumes.