  The main goal of this paper is to define a certain Chow weight structure
$w_{Chow}$ on the category $DM_c(S)$ of (constructible) $cdh$-motives over an
equicharacteristic scheme $S$. In contrast to the previous papers of D.
H\'ebert and the first author on weights for relative motives (with rational
coefficients), we can achieve our goal for motives with integral coefficients
(if $\operatorname{char}S=0$; if $\operatorname{char}S=p>0$ then we consider
motives with $\mathbb{Z}[\frac{1}{p}]$-coefficients). We prove that the
properties of the Chow weight structures that were previously established for
$\mathbb{Q}$-linear motives can be carried over to this "integral" context (and
we generalize some of them using certain new methods). In this paper we mostly
study the version of $w_{Chow}$ defined via "gluing from strata"; this enables
us to define Chow weight structures for a wide class of base schemes.
  As a consequence, we certainly obtain certain (Chow)-weight spectral
sequences and filtrations for any (co)homology of motives.
