Let $F:C \to D$ be a full and faithful functor between small categories. Then we get a triple of adjoint functors $F_! \dashv F^* \dashv F_*$, with $$F_!:Set^{C^{op}} \to Set^{D^{op}}.$$

Notice that $F_!=Lan_{y_C} \left(y_D \circ F\right),$ where in both cases $y$ denotes the Yoneda embedding, so that $F_!$ is left-exact if and only if $y_D \circ F$ is filtering. (Remark: I do NOT want to assume that C has finite limits, since it doesn't in my example,so filtering $\ne$ left-exact).

I'm looking for a stronger statement however. Suppose that $y_D \circ F$ is NOT filtering, so that $F_!$ is NOT left-exact. Nonetheless, $F_!$ may preserve certain finite limits (perhaps those in the image of a certain left-exact functor etc.). My question is, can one characterize (or give a sufficient condition for) those limits in $Set^{C^{op}}$ which ARE preserved by $F_!$?