  We adopt a non-perturbative relativistic constituent-quark model for the
pi-meson electromagnetic form factor, which have successfully predicted
experimental results, and supplement it with the effective momentum-dependent
quark mass to study quantitatively the transition to the perturbative QCD
asymptotics. The required asymptotical behaviour (including both the Q^{-2}
fall-off and the correct coefficient) settles down automatically when the quark
mass is switched off; however, the present experimental data on the form factor
suggest that this cannot happen at the values of the momentum transfer below
~10 GeV^2. The effective constituent-quark mass below this scale acquires
substantial non-perturbative contributions.
