Let L F be a finite separable field extension of degree n, X a smooth quasi-projective L-scheme, and R ( X) the Weil transfer of X with respect to L F . The map Z R (Z) of the set of simple cycles Z ⊂ X extends in a natural way to a map Z (X) → Z ( R (X)) on the whole group of algebraic cycles Z ( X). This map factors through the rational equivalence of cycles and induces this way a map of the Chow groups CH( X) → CH( R (X)), which, in its turn, produces a natural functor of the categories of Chow correspondences CV (L) → CV (F). Restricting to the graded components, one has a map Z ∗(X) → Z n · ∗( R (X)), which produces a functor of the categories of degree 0 Chow correspondences CV 0(L) → CV 0(F), a functor of the categories of the Grothendieck Chow-motives M (L) → M (F), as well as functors of several other classical motivic categories.