  Building on previous works by Bilu, Chambert-Loir and Loeser, we study the
asymptotic behaviour of the moduli space of sections of a given family over a
smooth projective curve, assuming that the generic fiber is an equivariant
compactification of a finite dimensional vector space. Working in a suitable
Grothendieck ring of varieties, we show that the class of these moduli spaces
converges, modulo an adequate normalisation, to a non-zero effective element,
when the class of the sections goes arbitrary far from the boundary of the dual
of the effective cone. The limit can be interpreted as a motivic Euler product
in the sense of Bilu's thesis. This result provides a positive answer to a
motivic version of the Batyrev-Manin-Peyre conjectures in this particular
setting.
