Zero-cycles on surfaces dominated by products of hyperelliptic curves

Conditionally on the finiteness of the relevant Tate-Shafarevich groups, we prove a local-to-global result for zero-cycles of degree $1$ on the surfaces given by $y^2=f_1(x_1)f_2(x_2)$, where the polynomials $f_1$ and $f_2$ are algebraically general. The proof combines the fibration method, parity results for $2$-Selmer groups in quadratic twist families, and a variant of a theorem of Morgan on the variation of the Cassels-Tate pairing.