  Let $k$ be an algebraically closed field of characteristic $p>0$ and let
$\mathbb{F}$ be an algebraically closed field of characteristic $0$. Recently,
together with Bouc, we introduced the notion of functorial equivalences between
blocks of finite groups and proved that given a $p$-group $D$, there is only a
finite number of pairs $(G,b)$ of a finite group $G$ and a block $b$ of $kG$
with defect groups isomorphic to $D$, up to functorial equivalence over
$\mathbb{F}$. In this paper, we classify the functorial equivalence classes
over $\mathbb{F}$ of blocks with cyclic defect groups and $2$-blocks of defects
$2$ and $3$. In particular, we prove that for all these blocks, the functorial
equivalence classes depend only on the fusion system of the block.
