  We study the category $DM(S)$ of Beilinson motives (as described by Cisinski
and Deglise) over a more or less general base scheme $S$, and establish several
nice properties for a version $t_{hom}(S)$ of the perverse homotopy
$t$-structure (essentially defined by Ayoub) for it. $t_{hom}(S)$ is
characterized in terms of certain stalks of an $S$-motif $H$ and its Tate
twists at fields over $S$; it is closely related to certain coniveau spectral
sequences for the cohomology of (the Borel-Moore motives of) arbitrary finite
type $S$-schemes.
  We conjecture that the heart of $t_{hom}(S)$ is given by cycle modules over
$S$ (as defined by Rost); for varieties over characteristic $0$ fields this
conjecture was recently proved by Deglise. Our definition of $t_{hom}(S)$ is
closely related to a new effectivity filtration for $DM(S)$ (and for the
subcategory of Chow $S$-motives in it). We also sketch the construction of a
certain Gersten weight structure for the category of $S$-comotives; this weight
structure yields one more description of $t_{hom}(S)$ and its heart.
