  We report progress on the determination and study of quarkonium production
within the fragmentation approximation. Our analyses address the moderate and
large transverse-momentum regime, where the collinear fragmentation of a single
parton is expected to dominate over the short-distance production, directly
from the hard scattering, of the constituent $(Q \bar{Q})$ system. Parton
fragmentation channels to pseudoscalar and vector quarkonia are built on the
basis of non-Relativistic QCD next-to-leading computations, which we use to
model initial-scale fragmentation inputs. Thus, a preliminary family of
Variable-Flavor Number-Scheme (VFNS) fragmentation functions, named NRFF1.0,
are constructed through standard DGLAP evolution. Statistical uncertainties are
obtained from a Monte Carlo, replica-like approach embodying missing
higher-order uncertainties.
