A dynamical empirical model of the near-geosynchronous magnetic field has been constructed, based on a recently developed radial basis functions approach and a multiyear set of spacecraft data taken by Time History of Events and Macroscale Interactions during Substorms (THEMIS), Polar, Cluster, and Van Allen Probes missions including 133 geomagnetic storms in the time interval between 1996 and 2016. The model describes the field as a function of Cartesian solar magnetic coordinates, dipole tilt angle, and solar wind ram pressure and of a set of dynamic variables representing the response of the magnetosphere to the external driving/loading during the active phase of a space weather event, followed by the internal relaxation/dissipation during the storm recovery. In terms of the disturbance level, the model's validity range extends to intense storms with peak SYM-H values down to -150nT. The spatial validity domain is a toroidal volume bounded by the inner (L similar to 5) and outer (L similar to 9) dipolar L shells, which allows the model to be used for tracing field lines to magnetically map geosynchronous spacecraft locations down to low altitudes. The model has been validated on independent out-of-sample magnetic field data and compared with an earlier empirical model and GOES 15 data taken in 2012 and 2015.