Functoriality of HKR isomorphisms

For a closed embedding of smooth schemes $X\hookrightarrow S$ with a fixed first order splitting, one can construct HKR isomorphisms between the derived scheme $X{\times }_S^{R}X$ and the total space of the shifted normal bundle ${\mathbb{N}}_{X\mathrm{/}S}[-1]$, due to Arinkin-C\u{a}ld\u{a}raru, Arinkin-C\u{a}ld\u{a}raru-Hablicsek, and Grivaux. In this paper, we study functoriality property of the HKR isomorphisms for a sequence of closed embeddings $X\hookrightarrow Y\hookrightarrow S$. The HKR isomorphism is functorial when a certain cohomology class, which we call the Bass-Quillen class, vanishes. We obtain Lie theoretic interpretations for the HKR isomorphisms and for the Bass-Quillen class as well.