  We investigate the Lyubeznik numbers, and the injective dimension of local
cohomology modules, of finitely generated $\mathbb{Z}$-algebras. We prove that
the mixed characteristic Lyubeznik numbers and the standard ones agree locally
for almost all reductions to positive characteristic. Additionally, we address
an open question of Lyubeznik that asks whether the injective dimension of a
local cohomology module over a regular ring is bounded above by the dimension
of its support. Although we show that the answer is affirmative for several
families of $\mathbb{Z}$-algebras, we also exhibit an example where this bound
fails to hold. This example settles Lyubeznik's question, and illustrates one
way that the behavior of local cohomology modules of regular rings of equal
characteristic and of mixed characteristic can differ.
