  We study certain triangulated categories of $K$-motives $DK(-)$ over a wide
class of base schemes, and define certain "weights" for them. We relate the
weights of particular $K$-motives to (negative) homotopy invariant $K$-groups
(tensored by $\mathbb{Z}[S^{-1}]$ for $S$ being the set of "non-invertible
primes") $\mathcal{K}_*(-)$. Our results yield a (new) result on the vanishing
of $\mathcal{K}_i(-)$ and of relative $\mathcal{K}$-groups for $i$ being "too
negative"; this statement is closely related to a question of Ch. Weibel. We
also prove that $\mathcal{K}_i(-)$ for $i<0$ is "supported in codimension
$-i$". Moreover, we establish several criteria for bounding (below) the weights
of $f_*(1_Y)$ ($1_Y$ is the tensor unit in $DK(Y)$); this automatically implies
the vanishing of the corresponding $E_2$-terms of Chow-weight spectral
sequences (and of the factors of the corresponding Chow-weight filtrations) for
any (co)homology of these motives. Our methods of bounding weights can be
applied to various "motivic" triangulated categories; this yields some new
statements on (constructible) complexes of \'etale sheaves.
  We also relate the weights of $K$-motives with rational coefficients to that
of Beilinson motives; the Chow-weight spectral sequences converging to their
$\mathbb{Q}_l$-\'etale (co)homology yield Deligne-type weights for the latter.
Somewhat surprisingly, we are able to prove in certain ("extreme") cases that
the corresponding weight bounds coming from \'etale (co)homology are precise.
We illustrate these statements by simple examples.
