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We study the category $DM(S)$ of Beilinson motives (as described by Cisinskiand Deglise) over a more or less general base scheme $S$, and establish severalnice properties for a version $t_{hom}(S)$ of the perverse homotopy$t$-structure (essentially defined by Ayoub) for it. $t_{hom}(S)$ ischaracterized in terms of certain stalks of an $S$-motif $H$ and its Tatetwists at fields over $S$; it is closely related to certain coniveau spectralsequences for the cohomology of (the Borel-Moore motives of) arbitrary finitetype $S$-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 thisconjecture was recently proved by Deglise. Our definition of $t_{hom}(S)$ isclosely related to a new effectivity filtration for $DM(S)$ (and for thesubcategory of Chow $S$-motives in it). We also sketch the construction of acertain Gersten weight structure for the category of $S$-comotives; this weightstructure yields one more description of $t_{hom}(S)$ and its heart.

On perverse homotopy $t$-structures, coniveau spectral sequences, cycle modules, and relative Gersten weight structures