The paper is aimed at defining 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. Hébert and the first author on weights for relative motives (with rational coefficients), this goal is achieved for motives with integral coefficients (if char S = 0; if char S = p > 0, then motives with ℤ[1/p]-coefficients are considered). It is proved that the properties of the Chow weight structures that were previously established for Q-linear motives can be carried over to this "integral" context (and some of them are generalized using certain new methods). Mostly, the version of w_Chow defined via "gluing from strata" is studied; this makes it possible to define Chow weight structures for a wide class of base schemes. As a consequence, certain (Chow)-weight spectral sequences and filtrations are obtained on any (co)homology of motives.