In the third paragraph of this paper it's claimed that a submersion admitting a complete Ehresmann connection is a fiber bundle. The proof sketch says to reduce to the base space an open ball in Euclidean space, and then perform parallel transport along radial segments.
Since the connection is complete, any radial segment lifts to a curve with horizontal velocity field. Flowing along the velocity field gives parallel transport along a single radial segment. To construct a trivialization however, I need a global horizontal field (flowing along it is the trivialization). I don't see how to combine the individual horizontal fields over radial segments into a horizontal vector field over the entire ball.
I'm sure the underlying principle here is smooth dependence of ODE on parameters, but I'm struggling to write things down. The only ODE that comes to mind is that of parallel transport given by the covariant derivative induced by the Ehresmann connection, but I just can't compile things.