We study quasiparticle transport in disordered quasi-one-dimensional superconductors of symmetry class D in the presence of m protected topological channels. Such systems can arise at the junction between a pair of two-dimensional topological superconductors with different topological numbers. We calculate conductance variance due to mesoscopic fluctuations and shot noise power with the help of n = 2 (two-replica) nonlinear supersymmetric sigma-model, with protected channels taken into account by a Wess-Zumino-Witten (WZW) topological term in the action. The calculation requires Fourier analysis on the sigma-model supermanifold with a non-trivial connection stemming from the WZW term. Eigenfunctions of the corresponding Laplace-Beltrami operator were constructed with the help of Iwasawa parametrisation. The presence of the WZW term leads to the emergence of additional families of eigenfunctions compared to m = 0 case.