  In a series of papers, we present a comprehensive analytic study of the
global motion of growing dust grains in protoplanetary discs, addressing both
the radial drift and the vertical settling of the particles. Here we study how
the radial drift of dust particles is affected by grain growth. In a first
step, toy models in which grain growth can either be constant, accelerate or
decelerate are introduced. The equations of motion are analytically integrable
and therefore the grains dynamics is easy to understand.
  The radial motion of growing grains is governed by the relative efficiency of
the growth and migration processes which is expressed by the dimensionless
parameter Lambda, as well as the exponents for the gas surface density and
temperature profiles, denoted p and q respectively. When Lambda is of order
unity, growth and migration are strongly coupled, providing the most efficient
radial drift. For the toy models considered, grains pile up when -p+q+1/2<0.
Importantly, we show the existence of a second process which can help discs to
retain their solid materials. For accelerating growth, grains end up their
migration at a finite radius, thus avoiding being accreted onto the central
star.
