Vertical flows and a general currential homotopy formula Florentiu Cibotaru 58A2549Q15transgression formulasMorse-Bott-Smale vector fieldsChern-Gauss-Bonnet Theoremsuperconnections 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 [D. Cibotaru, \textit{Vertical Morse-Bott-Smale flows and characteristic forms}, submitted] contains several applications, one of which is an answer to a question of Quillen. We also prove a Poincar\'e 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. Indiana University Mathematics Journal 2016 text pdf 10.1512/iumj.2016.65.5762 10.1512/iumj.2016.65.5762 en Indiana Univ. Math. J. 65 (2016) 93 - 169 state-of-the-art mathematics http://iumj.org/access/