This may not be precisely what you want, but a function field analogue of Beilinson's conjectures is formulated in R. Sreekantan, Non-Archimedean regulator maps and special values of $L$-functions, Cycles, motives and Shimura varieties, 469–492. Tata Inst. Fund. Res. Stud. Math., 21. Published for the Tata Institute of Fundamental Research, Mumbai, 2010.

Note as well that in W. Raskind, Higher l-adic Abel-Jacobi mappings and filtrations on Chow groups, Duke Math. J.78(1995), no.1, 33–57 it is shown that if $X$ is a smooth projective variety of dimension $n$ over a function field $k$ in one variable over a finite field of characteristic $p\neq\ell$, and such that $X$ has a proper regular model over $\mathcal{O}_{k}$. Then the $\ell$-adic Abel-Jacobi map $d_{2}^{n}:\mathrm{Fil}^{2}\mathrm{CH}^{n}(X)\otimes\mathbb{Q}\rightarrow H^{2}(k,H^{2n-2}(\overline{X},\mathbb{Q}_{\ell}(i)))$ is zero.

This may not be precisely what you want, but a function field analogue of Beilinson's conjectures is formulated in R. Sreekantan, Non-Archimedean regulator maps and special values of $L$-functions, Cycles, motives and Shimura varieties, 469–492. Tata Inst. Fund. Res. Stud. Math., 21. Published for the Tata Institute of Fundamental Research, Mumbai, 2010.

Note as well that in W. Raskind, Higher l-adic Abel-Jacobi mappings and filtrations on Chow groups, Duke Math. J.78(1995), no.1, 33–57 it is shown that if $X$ is a smooth projective variety of dimension $n$ over a function field $k$ in one variable over a finite field of characteristic $p\neq\ell$, and such that $X$ has a proper regular model over $\mathcal{O}_{k}$, then the $\ell$-adic Abel-Jacobi map $d_{2}^{n}:\mathrm{Fil}^{2}\mathrm{CH}^{n}(X)\otimes\mathbb{Q}\rightarrow H^{2}(k,H^{2n-2}(\overline{X},\mathbb{Q}_{\ell}(i)))$ is zero.