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, and certainly there exists a morphism between their $k$-points (at the level of sets) since the Albanese map $CH_0(X)^0\to Alb(X)(k)$ is a bijection (well condidtional on Bloch-Beilinson for $k=\overline{\mathbb{Q}}$).