  The unpolarized twist-2 (leading) and twist-3 (subleading), T-even,
transverse-momentum dependent quark distributions in the pion are evaluated for
the first time by using the actual solution of a dynamical equation in
Minkowski space. The adopted theoretical framework is based on the homogeneous
Bethe-Salpeter integral equation with an interaction kernel given by a
one-gluon exchange, featuring an extended quark-gluon vertex. The masses of
quark and gluon as well as the interaction-vertex scale have been chosen in a
range suggested by lattice-QCD calculations, and calibrated to reproduce both
pion mass and decay constant. The sum rules to be fulfilled by the
transverse-momentum dependent distributions are carefully investigated,
particularly the leading-twist one, that has to match the collinear parton
distribution function, and hence can be scrutinized in terms of existing data
as well as theoretical predictions. Noteworthy, the joint use of the Fock
expansion of the pion state facilitates a more in-depth analysis of the content
of the pion Bethe-Salpeter amplitude, allowing for the first time to determine
the gluon contribution to the quark average longitudinal fraction, that results
to be $\sim 6\%$. The current analysis highlights the role of the gluon
exchanges through quantitative analysis of collinear and transverse-momentum
distributions, showing, e.g. for both leading and subleading-twists, an early
departure from the widely adopted exponential fall-off, for $|k_\perp|^2 >
m^2$, with the quark mass $\sim \Lambda_{QCD}$.
