Vertical Flows and a General Currential Homotopy Formula

We generalize some results of Harvey, Lawson, and Latschev about transgression formulas. The focus here is on flowing forms via vertical vector fields, especially tame Morse-Bott-Smale vector fields. We prove a general transgression formula including also a version covering non-compact situations. A second, companion paper [10] contains several applications, one of which is an answer to a question of Quillen. We also prove a Poincaré duality result concerning the trangression classes induced by the Pfaffian, construct the Maslov spark, give a short proof of the Chern-Gauss-Bonnet theorem, and re-prove a result of Getzler.