  For an inner form $\mathrm{G}$ of a general linear group or classical group
over a non-archimedean local field of odd residue characteristic, we decompose
the category of smooth representations on
$\mathbb{Z}[\mu_{p^{\infty}},1/p]$-modules by endo-parameter. We prove that
parabolic induction preserves these decompositions, and hence that it preserves
endo-parameters. Moreover, we show that the decomposition by endo-parameter is
the $\overline{\mathbb{Z}}[1/p]$-block decomposition; and, for $\mathrm{R}$ an
integral domain, introduce a graph whose connected components parameterize the
$\mathrm{R}$-blocks, in particular including the cases
$\mathrm{R}=\overline{\mathbb{Z}}_{\ell}$ and
$\mathrm{R}=\overline{\mathbb{F}}_\ell$ for $\ell\neq p$. From our description,
we deduce that the $\overline{\mathbb{Z}_\ell}$-blocks and
$\overline{\mathbb{F}_\ell}$-blocks of $\mathrm{G}$ are in natural bijection,
as had long been expected. Our methods also apply to the trivial endo-parameter
(i.e., the depth zero subcategory) of any connected reductive $p$-adic group,
providing an alternative approach to results of Dat and Lanard in depth zero.
Finally, under a technical assumption (known for inner forms of general linear
groups) we reduce the $\mathrm{R}$-block decomposition of $\mathrm{G}$ to depth
zero.
