The temporal evolution of the propagation environment plays a central role in integrated sensing and communication (ISAC) systems. A slow-time evolution manifests as channel aging in communication links, while a fast-time one is associated with non-zero Doppler clutter. Nevertheless, the joint impact of these two phenomena on ISAC performance has been largely overlooked. This paper addresses this research gap in a network utilizing orthogonal frequency division multiplexing waveforms. Here, a base station simultaneously serves a user equipment (UE) device and performs monostatic sensing. Channel aging is captured through an autoregressive model with exponential correlation decay. Clutter is modeled as a collection of uncorrelated, coherent patches with non-zero Doppler, resulting in a Kronecker-separable covariance structure. We propose an aging-aware channel estimator that uses prior pilot observations to estimate the time-varying UE channel, characterized by a non-isotropic multipath fading structure. The clutter's structure enables a novel low-complexity pre-detection radar processing pipeline: clutter statistics are estimated from raw data and subsequently used to suppress the clutter's action, after which range-angle and range-velocity maps are computed. We evaluate the influence of frame length and pilot history on channel estimation accuracy and demonstrate substantial performance gains over block fading in low-to-moderate mobility regimes. The sensing pipeline is implemented in a clutter-dominated environment, demonstrating that effective clutter suppression can be achieved under practical configurations. We analyze the robustness of our proposed pipeline against non-separable clutter by introducing a controllable degree of non-separability. Our results highlight the benefit of sensing streams and that our pipeline can withstand a moderate degree of non-separability.