A short and not very useful answer is that the limit doesn't exist in general. A longer and more useful answer is that it exists under some conditions, namely if the wave front of the distribution $S$ is transverse to the map $F$ (some explanations below).
This kind of problems belong to the microlocal analysis. A similar problem is whether/when we can multiply two distributions: again, we can multiply a distribution by a smooth function, and any distribution is the limit of a sequence of smooth functions. The condition is again formulated in terms of the wave front sets (namely that their sum should not contain any zero covector).
Now some details: every distribution $S$ on $M$ has its singular support $sing(S)\subset M$, away from which $S$ is a smooth function. The wave front of a distribution is a more detailed notion: $WF(S)$ is a (conical) subset of $T^*M$; one takes care not only of the singular points, but also of the (co)directions in which $S$ is singular at the point (see the wikipedia link for details). [The projection of $WF(S)$ to $M$ is, as you'd guess, $sing(S)$.]
If $F:N\to M$ is a smooth map then the composition $S\circ F$ (more often called "the pullback of $S$ by $F$", denoted $F^*S$) is defined if $F$ is transverse to $WF(S)$, which means that $F^*(\lambda)\neq 0$ for every $\lambda\in WF(S)$. Likewise, $S_1S_2$ is defined if $WF(S_1)+WF(S_2)\subset T^*M$ doesn't contain the zero covector at any point of $M$. [A little 'hint': if $F^*S$ exists then $WF(F^*S)\subset F^*WF(S)$, likewise $WF(S_1S_2)\subset WF(S_1)+WF(S_2)$; wave fronts contain (by their definition) no zero covectors.]