Metric mixed approximation along divisibility chains

Let $1=d_0<d_1<\cdots$ be a divisibility chain, $|\cdot|_{\mathcal D}$ the associated pseudo-absolute value, and $L(x)=\#\{j:d_j\leq x\}$. For every integer $a$, rational $\gamma$ and non-increasing $\psi$ we prove that $|n-a|_{\mathcal D}\|n\alpha-\gamma\|<\psi(n)$ has infinitely many solutions for almost every $\alpha$ if and only if $\sum_n\psi(n)L(n)$ diverges. For $a=\gamma=0$ this removes the averaged totient hypothesis in the theorem of Harrap and Haynes and answers their question; for $a\ne0$ the same holds for every real $\gamma$ satisfying the weak inhomogeneous Duffin--Schaeffer property. Together these results prove the conjecture of Chow and Jones on mixed inhomogeneous Gallagher sums except when $a=0$ and $\gamma$ is irrational. The case $a=0$ uses reduced fractions attached to least common multiples with a sparse set of chain elements. The case $a\ne0$ uses a counting theorem for arbitrary moving boxes on compatible residue classes; we show that the exponent in the associated gcd bound is optimal, extend the theorem to subpolynomial moduli, classify the fixed targets in finite products of $p$-adic integers for which divergence of the volume sum suffices, and deduce the Hausdorff-measure analogue.