We consider a general way to obtain Prékopa-Leindler and Borell-Brascamp-Lieb type inequalities from Brunn-Minkowski type inequalities and provide numerous examples. We use the same heuristic to prove a discrete version of the Prékopa-Leindler and Borell-Brascamp-Lieb inequalities for functions over $\mathbb{Z}^d$. These are the functional extensions of the discrete Brunn-Minkowski inequality conjectured by Ruzsa and recently established by Keevash, Tiba, and the author.