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

Motivic distribution of rational curves and twisted products of toric varieties