For a smooth projective scheme Y over W(k) we consider an element in the motivic Chow group of the reduction Y_{m} over the truncated Witt ring W_{m}(k) and give a “Hodge” criterion – using the crystalline cycle class in relative crystalline cohomology – for the element to lift to the continuous Chow group of the associated p -adic formal scheme Y_\bullet . The result extends previous work of Bloch–Esnault–Kerz on the p -adic variational Hodge conjecture to a relative setting. In the course of the proof we derive two new results on the relative de Rham–Witt complex and its Nygaard filtration, and work with a relative version of syntomic complexes to define relative motivic complexes for a smooth lifting of Y_{m} over the ind-scheme \mathrm{Spec}\,W_{\bullet}(W_{m}(k)) .