Metric mixed approximation along divisibility chains

Let 1=d₀<d₁<⋯ be a divisibility chain, |·|D the associated pseudo-absolute value, and L(x)=#:dⱼ≤ x\. For every integer a, rational γ and non-increasing ψ we prove that |n-a|D\|nα-γ\|<ψ(n) has infinitely many solutions for almost every α if and only if ∑ₙψ(n)L(n) diverges. For a=γ=0 this removes the averaged totient hypothesis in the theorem of Harrap and Haynes and answers their question; for a≠0 the same holds for every real γ 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 γ is irrational. The case $a=0$ uses reduced fractions attached to least common multiples with a sparse set of chain elements. The case a≠0 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.