Understanding the linear growth of disturbances due to external forcing is crucial for flow stability analysis, flow control, and uncertainty quantification. These applications typically require a large number of forward simulations of the forced linearized dynamics, often in a brute-force fashion. When dealing with simple steady-state or periodic base flows, there exist powerful and cost-effective solution operator techniques. Once constructed, these operators can be used to determine the response to various forcings with negligible computational cost. However, these methods are not applicable to problems with arbitrarily time-dependent base flows. This paper develops and investigates reduced-order modeling with time-dependent bases (TDBs) to build low-rank solution operators for forced linearized dynamics with arbitrarily time-dependent base flows. In particular, we use forced optimally time-dependent decomposition (f-OTD), which extracts the time-dependent correlated structures of the flow response to various excitations. We also demonstrate that in the case of a steady-state mean flow subject to harmonic forcing, the f-OTD subspace converges to the dominant resolvent analysis modes. The demonstration includes four cases: a toy model, the Burgers equation, the 2D temporally evolving jet, and two-dimensional decaying isotropic turbulence. In these cases, we demonstrate the utility of the low-rank operator for (i) identifying the excitation that leads to maximum amplification, and (ii) reconstructing the full-state flow without incurring additional cost.