A hydrodynamic description for inelastic Maxwell mixtures driven by a stochastic bath with friction is derived. Contrary to previous works where constitutive relations for the fluxes were restricted to states near the homogeneous steady state, here the set of Boltzmann kinetic equations is solved by means of the Chapman--Enskog method by considering a more general time-dependent reference state. Due to this choice, the transport coefficients are given in terms of the solutions of a set of nonlinear differential equations which must be in general numerically solved. The solution to these equations gives the transport coefficients in terms of the parameters of the mixture (masses, diameters, concentration, and coefficients of restitution) and the time-dependent (scaled) parameter $xi^*$ which determines the influence of the thermostat on the system. The Navier--Stokes transport coefficients are exactly obtained in the special cases of undriven mixtures ($xi^*=0$) and driven mixtures under steady conditions ($xi^*=xi_text{st}^*$, where $xi_text{st}^*$ is the value of the reduced noise strength at the steady state). As a complement, the results for inelastic Maxwell models (IMM) in both undriven and driven steady states are compared against approximate results for inelastic hard spheres (IHS) [Khalil and Garzo, Phys. Rev. E textbf{88}, 052201 (2013)]. While the IMM predictions for the diffusion transport coefficients show an excellent agreement with those derived for IHS, significant quantitative differences are specially found in the case of the heat flux transport coefficients.
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