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Abstract: Koopman theory has become widely used for predictions and analysis of nonlinear systems. Especially, kernel extended dynamic mode decomposition (kEDMD) embeds nonlinear dynamics in a reproducing-kernel Hilbert space. However, its Gram matrix grows with every snapshot, blocking online use when parameters drift. We present a low-rank approximation-based kEDMD algorithm that swaps out arbitrary old snapshots, inserts new ones, and updates the eigendecomposition with rank-two corrections, keeping both memory and cost fixed. The method efficiently tracks evolving systems on the van der Pol oscillator and adapts to unseen data and slow coefficient changes. In addition, it efficiently tracks a 50-point discretized Korteweg-de Vries flow and preserves predictive accuracy.