Anderson's $t$-motives satisfy most of what is expected of a reasonable category of mixed motives, except of course that everything is in positive characteristic. For instance, it is a linear category with a tensor product, we have a well behaved weight filtration, and there are realisation functors with good properties. (After all, the name "motive" was not chosen at random.)

However, a classical motive is something that one wants to associate with a scheme or variety, thinking of it as its universal cohomology, and now it bugs me that I have no geometric objects at hand to which I could associate $t$-motives. I don't even know whether these, if any, should be varieties or something else.

To make this a halfway real question: Let $C$ be a smooth proper curve over $\mathbb{F}_p(t)$ of genus $g$. Is there a natural, functorial way of associating with $C$ a pure $t$-motive $M(C)$ of rank $g$, in such a way of course that the cohomology of $C$ is related, via a functor morphism, to the realisations of $M(C)$? So I'm asking here for some kind of Jacobian construction.

But as I said, I don't even know whether varieties over $\mathbb{F}_p(t)$ are the right geometric objects to look at.