We propose the 'flow of a proof' as a construct that relates to the balance lecturers achieve in their presentation between the proof as a whole and its different parts, taking into account various aspects (informal, contextual) of proof classroom presentation. We conducted expert interviews in which we presented two different proofs of the same theorem to five experienced tertiary level mathematics lecturers and asked them to reflect on features of these proofs and of proofs in general, and on their considerations and practices when teaching proofs. The lecturers related to a rich collection of proof features and pedagogical considerations. Our findings demonstrate the subjectivity of lecturers' considerations as well as tensions they experience between striving towards mathematical accuracy and adjusting the proof presentation to their students; these tensions influence the flow of the proof, which is an outcome of the choices that the lecturers make. We discuss our findings in light of existing literature about mathematicians' pedagogical considerations and the existence of a shared pedagogical culture.