  We present a factorization theorem of the partonic Drell-Yan off-diagonal
processes $g\bar{q}\,(qg) \to \gamma^* + X$ in the kinematic threshold regime
$z=Q^2/\hat{s} \to 1$ at general subleading powers in the $(1-z)$ expansion.
Focusing on the first order of the expansion (next-to-leading power accuracy
with respect to the leading power $q \bar{q}$ channel), we validate the bare
factorization formula up to $\mathcal{O}(\alpha^2_s)$. This is achieved by
carrying out an explicit calculation of the generalized soft function in
$d$-dimensions using the reduction to master integrals and the differential
equations method. The collinear function is a universal object which we compute
from an operator matching equation at one-loop level. Next, we integrate the
soft and collinear functions over the convolution variables and remove the
remaining initial state collinear singularities through PDF renormalization.
The resulting expression agrees with the known cross section in the literature.
