I am not sure about using the LTP to find conditional CDF. For example, suppose $X\sim$ some distribution with a random parameter $\lambda\sim$ some discrete distribution. Then I have the following:
$p_X(x) = \sum_{k = 1}^{n} p(X\cap \lambda_k) = p(x|\lambda_{1})p(\lambda_{1}) + \ldots + p(x|\lambda_{n})P(\lambda_{n}) = \sum_{y\in S_\lambda}p(x | \lambda_k = y ) p({\lambda_k}) = F_{X|\lambda=y}(x)$
since $p_{\lambda_k} (y) = f(c)$, a constant i.e. a probability, thus $\sum_{y\in S_\lambda}p(x | \lambda_k = y ) f(c) = F_{X|\lambda=y}(x)$.
$\textbf{Questions:}$
Can we have $p_X(x) \stackrel{?}{=} F_{X|Y}(x)$ as shown above?
If the above is wrong, what is the general way of deriving CDF and PDF/PMF from LTP?