  We generalize the motivic incarnation morphism from the theory of arithmetic
integration to the relative case, where we work over a base variety S over a
field k of characteristic zero. We develop a theory of constructible effective
Chow motives over S, and we show how to associate a motive to any S-variety.
  We give a geometric proof of relative quantifier elimination for
pseudo-finite fields, and we construct a morphism from the Grothendieck ring of
the theory of pseudo-finite fields over S, to the tensor product of Q with the
Grothendieck ring of constructible effective Chow motives. This morphism yields
a motivic realization for the parametrized arithmetic motivic integrals of
Cluckers-Loeser.
  Finally, we define relative arc and jet spaces, and the three relative
motivic generating series.
