A healthy body of evidence says that birational geometry and derived categories are intimately bound. Even so, many basic questions are still open. One of the most central questions is the conjecture of Bondal and Orlov (later extended by Kawamata) that says two smooth projective varieties related by a flop are actually derived equivalent. The first step in resolving this question is understanding how to produce functors from rational maps. In work with Diemer and Favero, we provided a method to construct an integral kernel associated to any D-flip of normal varieties with Q-Cartier D. Conjecturally, this can be used to answer Bondal and Orlov's question. In this talk, we will discuss the construction and natural extensions of it. In particular, we will highlight work with Chidambaram, Favero, McFaddin, and Vandermolen relating to the what has been termed a Grassmann flop.