  We present explicit formulas for total crossing events in the multi-species
asymmetric exclusion process ($r$-ASEP) with underlying
$U_q(\widehat{\mathfrak{sl}}_{r+1})$ symmetry. In the case of the two-species
TASEP these can be derived using an explicit expression for the general
transition probability on $\mathbb{Z}$ in terms of a multiple contour integral
derived from a nested Bethe ansatz approach. For the general $r$-ASEP we employ
a vertex model approach within which the probability of total crossing can be
derived from partial symmetrization of an explicit high rank rainbow partition
function. In the case of $r$-TASEP, the total crossing probability can be show
to reduce to a multiple integral over the product of $r$ determinants. For
$2$-TASEP we additionally derive convenient formulas for cumulative total
crossing probabilities using Bernoulli-step initial conditions for particles of
type 2 and type 1 respectively.
