dg.differential geometry – “Lie theory” for anchored bundles and reflexive graphs

Perhaps Lie theory is not the correct term, but I’m thinking of the intermediate result in the Lie groupoid to Lie algebroid correspondence. Given a Lie groupoid $G$ over $M$, we may construct the Lie algebroid of $G$ by taking the pullback of the cospan
$$ M xrightarrow{(e,0)} G times TM xleftarrow{(p, Ts)} TG$$
Now, if $G$ were just a reflexive, involutive graph rather than a groupoid, we would still get a vector bundle over $M$ from this pullback, with an anchor $varrho := A hookrightarrow TG xrightarrow{Tt} TM$.

Has anyone written down the analogue of “Lie integration” for this setting? If I move to synthetic differential geometry there’s a way through using enriched sketches, but I’d like to see something a little more down to earth.