A new solution technique is derived for the time-dependent transport equation. This approach extends the steady-state coarse-mesh transport method that is based on global-local decompositions of large (i.e. full-core) neutron transport problems. The new method is based on polynomial expansions of the space, angle and time variables in a response-based formulation of the transport equation. The local problem (coarse mesh) solutions, which are entirely decoupled from each other, are characterized by space-, angle- and time-dependent response functions. These response functions are, in turn, used to couple an arbitrary sequence of local problems to form the solution of a much larger global problem. In the current work, the local problem (response function) computations are performed using the Monte Carlo method, while the global (coupling) problem is solved deterministically. The spatial coupling is performed by orthogonal polynomial expansions of the partial currents on the local problem surfaces, and similarly, the timedependent response of the system (i.e. the time-varying flux) is computed by convolving the time-dependent surface partial currents and time-dependent volumetric sources against pre-computed time-dependent response kernels.