This paper develops a tractable analytical channel model for first-hitting-time molecular communication (MC) systems under time-varying drift. While existing studies of nonstationary transport rely primarily on numerical solutions of advection-diffusion equations or parametric impulse-response fitting, they do not provide an explicit analytical description of trajectory-level arrival dynamics at absorbing boundaries. By adopting a change-of-measure formulation, we reveal a structural decomposition of the first-hitting-time density into a cumulative-drift displacement term and a stochastic boundary-flux modulation factor. This leads to a closed-form Corrected-Inverse-Gaussian (C-IG) density that generalizes the classical IG model to nonstationary drift while preserving O(1) evaluation complexity. Monte Carlo simulations under both smooth pulsatile and abrupt switching drift profiles confirm that the proposed C-IG model accurately captures complex transport phenomena, including phase modulation, multi-pulse dispersion, and transient backflow -- effects that traditionally complicate symbol synchronization and induce severe inter-symbol interference. The resulting framework provides a physics-informed, computationally efficient channel model suitable for system-level analysis and advanced receiver design, such as real-time maximum likelihood detection, in dynamic biological and MC environments.

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