  For a connected reductive group $G$ defined over a non-archimedean local
field $F$, we consider the Bernstein blocks in the category of smooth
representations of $G(F)$. Bernstein blocks whose cuspidal support involves a
regular supercuspidal representation are called $\textit{regular}$ Bernstein
blocks. Most Bernstein blocks are regular when the residual characteristic of
$F$ is not too small. Under mild hypotheses on the residual characteristic, we
show that the Bernstein center of a regular Bernstein block of $G(F)$ is
isomorphic to the Bernstein center of a regular depth-zero Bernstein block of
$G^{0}(F)$, where $G^{0}$ is a certain twisted Levi subgroup of $G$. In some
cases, we show that the blocks themselves are equivalent, and as a consequence
we prove the ABPS Conjecture in some new cases.
