Let $X$ be some discrete random variable, i.e. $\mathbb P(X=x_j)=p_j$ for $x_j\in\mathbb R$ and $\sum_{j=1}^J p_j=1$. Furthermore, let $L_1,\ldots, L_n$ be a list of be random variables of which we only the conditional density function $$f_{L_i} (\ell \mid X=x)$$ with $\ell\in\mathbb R$.
My question is, can we say anything about the condiditional density $$f_L(\ell\mid X=x)$$ of $L:=\sum_{i=1}^n L_i$ ? That is, can we write $f_L$ in terms of $f_{L_i}$ ?
For example, ignoring the conditional part, assuming that the $L_i$ are independent and continuous, and setting $n=2$ gives us the convolution: $$f_L(\ell) = \int f_{L_1}(\ell-x) f_{L_2}(x)dx$$
Is there a similar formula for the case I described above, or generally speaking, is it at all possible to obtain $f_L(\ell\mid X=x)$ ?