  The goal of this paper is to prove: if certain 'standard' conjectures on
motives over algebraically closed fields hold, then over any 'reasonable' $S$
there exists a motivic $t$-structure for the category of Voevodsky's
$S$-motives (as constructed by Cisinski and Deglise). If $S$ is 'very
reasonable' (for example, of finite type over a field) then the heart of this
$t$-structure (the category of mixed motivic sheaves over $S$) is endowed with
a weight filtration with semi-simple factors. We also prove a certain 'motivic
decomposition theorem' (assuming the conjectures mentioned) and characterize
semi-simple motivic sheaves over $S$ in terms of those over its residue fields.
Our main tool is the theory of weight structures. We actually prove somewhat
more than the existence of a weight filtration for mixed motivic sheaves: we
prove that the motivic $t$-structure is transversal to the Chow weight
structure for $S$-motives (that was introduced previously and independently by
D. Hebert and the author; weight structures and their transversality with
t-structures were also defined by the author in recent papers). We also deduce
several properties of mixed motivic sheaves from this fact. Our reasoning
relies on the degeneration of Chow-weight spectral sequences for 'perverse
'etale homology' (that we prove unconditionally); this statement also yields
the existence of the Chow-weight filtration for such (co)homology that is
strictly restricted by ('motivic') morphisms.
