  Parton branching methods underlie the Monte Carlo (MC) generators, being
therefore of key importance for obtaining high energy physics predictions. We
construct a new parton branching algorithm which for the first time
incorporates the off-shell, transverse-momentum dependent (TMD) splitting
functions, defined from the high-energy limit of partonic decay amplitudes.
Based on these TMD splitting functions we construct a new TMD Sudakov form
factor. We present the first MC implementation of the algorithm for the
evolution of the TMD and integrated parton distribution functions (PDFs). We
use this implementation to evaluate small-$x$ corrections to the distributions
and to verify the momentum sum rule. The presented study is a first step
towards a full TMD MC generator covering the small-$x$ phase space.
