  We prove a necessary and sufficient criterion for the ring of integer-valued
polynomials to behave well under localization. Then, we study how the Picard
group of $\mathrm{Int}(D)$ and the quotient group
$\mathcal{P}(D):=\mathrm{Pic}(\mathrm{Int}(D))/\mathrm{Pic}(D)$ behave in
relation to Jaffard, weak Jaffard and pre-Jaffard families; in particular, we
show that $\mathcal{P}(D)\simeq\bigoplus\mathcal{P}(T)$ when $T$ ranges in a
Jaffard family of $D$, and study when similar isomorphisms hold when $T$ ranges
in a pre-Jaffard family. In particular, we show that the previous isomorphism
holds when $D$ is an almost Dedekind domain such that the ring integer-valued
polynomials behave well under localization and such that the maximal space of
$D$ is scattered with respect to the inverse topology.
