Databases recording cattle exchanges offer unique opportunities for a better understanding and fighting of disease spreading. Most studies model contacts with (sequences of) networks, but this approach neglects important dynamical features of exchanges, that are known to play a key role in spreading. We use here a fully dynamic modeling of contacts and empirically compare the spreading outbreaks obtained with it to the ones obtained with network approaches. We show that neglecting time information leads to significant overestimates of actual sizes of spreading cascades, and that these sizes are much more heterogeneous than generally assumed. Our approach also makes it possible to study the speed of spreading, and we show that the observed speeds vary greatly, even for a same cascade size.