  This work concerns asymptotical stabilisation phenomena occurring in the
moduli space of sections of certain algebraic families over a smooth projective
curve, whenever the generic fibre of the family is a smooth projective Fano
variety, or not far from being Fano.
  We describe the expected behaviour of the class, in a ring of motivic
integration, of the moduli space of sections of given numerical class. Up to an
adequate normalisation, it should converge, when the class of the sections goes
arbitrarily far from the boundary of the dual of the effective cone, to an
effective element given by a motivic Euler product. Such a principle can be
seen as an analogue for rational curves of the Batyrev-Manin-Peyre principle
for rational points.
  The central tool of this article is the property of equidistribution of
curves. We show that this notion does not depend on the choice of a model of
the generic fibre, and that equidistribution of curves holds for smooth
projective split toric varieties. As an application, we study the
Batyrev-Manin-Peyre principle for curves on a certain kind of twisted products.
