$\DeclareMathOperator\Alb{Alb}\DeclareMathOperator\CH{CH}$I'm wondering how the Albanese functor interacts with correspondences. Specifically if $X$, $Y$ are say smooth projective schemes over an algebraically closed field $k$ and $Z$ a correspondence between them. Does this induce a morphism of abelian varieties $$\Alb(X)\to \Alb(Y)?$$ 

If $X$, $Y$ are curves this is the case as the Albanese and Jacobian agree, then one can quote say Cor 6.3 of Milne's [notes on Jacobian Varieties][1], which says that morphisms on Jacobians of curves are in bijection to those divisors (or line bundles) which are trivial after pulling back to either factor.


  [1]: https://www.jmilne.org/math/xnotes/JVs.pdf