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This study introduces a new type of space-time autoregressive (STAR) filtering algorithm for space-time adaptive processing (STAP) operating in a clutter environment that is not strictly stationary in slow time. The original STAR approach based on stationary autoregressive (AR) model, despite enjoying a fast convergence rate, suffers significant performance degradation when dealing with non-stationary clutter processes. To remedy this, the new proposed algorithm invokes a 'relaxed' AR model, that is, the time-varying autoregressive (TVAR) model, and is called time-varying space-time autoregressive (TV-STAR) filtering. The authors demonstrate that, for stationary case, the two filters have identical output signal-to-interference plus noise ratio with known interference covariance, but the convergence rate of TV-STAR is somewhat inferior to STAR with finite sample support. However, in the non-stationary case, the STAR filter totally fails because of 'model-mismatch', whereas TV-STAR exhibits a commensurate performance with respect to the stationary case. Meanwhile, TV-STAR is shown to offer a favourable convergence rate over reduced-rank STAP techniques such as eigencanceler method in both cases.