  We propose a new procedure for parton-to-hadron fragmentation in
proton-proton collisions. The hadronization is considered in the parton-parton
center-of-mass system and hadrons are assumed to be emitted in the direction of
outgoing partons in that frame and then their four-momenta are transformed to
the overall center-of-mass system using relevant Lorentz transformations and
appropriate distributions are constructed. For heavy hadron production the
energy-momentum condition is imposed. In many cases our procedure disagrees
with the commonly used rule that rapidity of produced hadron is the same as
rapidity of the parent parton. We illustrate the newly proposed scheme for
production of $D$ mesons, $\Lambda_c$ baryon and $\eta_c$ quarkonium in
leading-order approach to parton production. The transverse momentum, rapidity
and Feynman-$x_F$ distributions are shown. We consider $c \to D$ and $g \to D$
as well as $c \to \eta_c$ and $g \to \eta_c$ hadronization processes. The
resulting rapidity distributions are much narrower than those obtained in the
traditional approach with parton-hadron rapidity equivalence. This has
consequences both for mid- and forward rapidities and could have consequences
for high-energy neutrino production in the Earth's atmosphere. We discuss also
$c \to \Lambda_c^+$ and $b \to \Lambda_c^+$ fragmentation and find that the
second mechanism could be partly responsible for the enhanced production of
$\Lambda_c^{\pm}$ observed recently by the ALICE collaboration.
