We study the asymptotic behaviour of random factorizations of the $n$-cycle into transpositions of fixed genus $g>0$. They have a geometric interpretation as branched covers of the sphere and their enumeration as Hurwitz numbers was extensively studied in algebraic combinatorics and enumerative geometry. On the probabilistic side, several models and properties of permutation factorizations were studied in previous works, in particular minimal factorizations of cycles into transpositions (which corresponds to the case $g=0$ of this work). Using the representation of factorizations via unicellular maps, we first exhibit an algorithm which samples an asymptotically uniform factorization of genus $g$ in linear time. In a second time, we code a factorization as a process of chords appearing one by one in the unit disk, and we prove the convergence (as $n\to\infty$) of the process associated with a uniform genus $g$ factorization of the $n$-cycle. The limit process can be explicitly constructed from a Brownian excursion. Finally, we establish the convergence of a natural genus process, coding the appearance of the successive genera in the factorization.