Cyclin‐dependent kinases (CDKs), the master regulators of cell division, are activated by different cyclins at different cell cycle stages. In addition to being activators of CDKs, cyclins recognize various linear motifs to target CDK activity to specific proteins. We uncovered a cyclin docking motif, NLxxxL, that contributes to phosphorylation‐dependent degradation of the CDK inhibitor Far1 at the G1/S stage in the yeast Saccharomyces cerevisiae. This motif is recognized exclusively by S‐phase CDK (S‐CDK) Clb5/6‐Cdc28 and is considerably more potent than the conventional RxL docking motif. The NLxxxL and RxL motifs were found to overlap in some target proteins, suggesting that cyclin docking motifs can evolve to switch from one to another for fine‐tuning of cell cycle events. Using time‐lapse fluorescence microscopy, we show how different docking connections temporally control phosphorylation‐driven target degradation. This also revealed a differential function of the phosphoadaptor protein Cks1, as Cks1 docking potentiated degron phosphorylation of RxL‐containing but not of NLxxxL‐containing substrates. The NLxxxL motif was found to govern S‐cyclin‐specificity in multiple yeast CDK targets including Fin1, Lif1, and Slx4, suggesting its wider importance.