Down-linking $(K_v,\Gamma)$-designs to $P_3$-designs

Let $\Gamma'$ be a subgraph of a graph $\Gamma$.
We define a \emph{down-link} from a $(K_v,\Gamma)$-design $\cB$ to a
$(K_n,\Gamma')$-design $\cB'$ as a map $f:\cB\to \cB'$
mapping any block of $\cB$ into one of its subgraphs.
This is a new concept, closely
related with both the notion of
\emph{metamorphosis} and that of \emph{embedding}.
In the present paper we study down-links in general and prove that any
$(K_v,\Gamma)$-design might be down-linked to a $(K_n,\Gamma')$-design,
provided that $n$ is admissible and large enough.
We also show that
if $\Gamma'=P_3$, it is always possible to find a down-link
to a design of order at most $v+3$.
This bound is then improved
for several classes of graphs $\Gamma$, by providing explicit
constructions.