  Let $G$ be a finite group and $(K,\mathcal{O},k)$ be a $p$-modular system.
Let $R=\mathcal{O}$ or $k$. There is a bijection between the blocks of the
group algebra and the blocks of the so-called $p$-local Mackey algebra
$\mu_{R}^{1}(G)$. Let $b$ be a block of $RG$ with abelian defect group $D$. Let
$b'$ be its Brauer correspondant in $N_{G}(D)$. It is conjectured by Brou\'e
that the blocks $RGb$ and $RN_{G}(D)b'$ are derived equivalent. Here we look at
equivalences between the corresponding blocks of $p$-local Mackey algebras. We
prove that an analogue of the Brou\'e's conjecture is true for the $p$-local
Mackey algebras in the following cases: for the principal blocks of
$p$-nilpotent groups and for blocks with defect $1$. We also point out the
probable importance of \emph{splendid} equivalences for the Mackey algebras.
