  We investigate long time asymptotics of the modified Camassa-Holm equation in
three transition zones under a nonzero background. The first transition zone
lies between the soliton region and the first oscillatory region, the second
one lies between the second oscillatory region and the fast decay region, and
possibly, the third one, namely, the collisionless shock region, that bridges
the first transition region and the first oscillatory region. Under a low
regularity condition on the initial data, we obtain Painlev\'e-type asymptotic
formulas in the first two transition regions, while the transient asymptotics
in the third region involves the Jacobi theta function. We establish our
results by performing a $\bar{\partial}$ nonlinear steepest descent analysis to
the associated Riemann-Hilbert problem.
