  We study positive characteristic multiple zeta values associated to general
curves over $\mathbb F_q$ together with an $\mathbb F_q$-rational point
$\infty$ as introduced by Thakur. For the case of the projective line these
values were defined as analogues of classical multiple zeta values. In the
present paper we first establish a general non-commutative factorization of
exponential series associated to certain lattices of rank one. Next we
introduce universal families of multiple zeta values of Thakur and show that
they are Eulerian in full generality. In particular, we prove a conjecture of
Lara Rodriguez and Thakur arXiv:2003.12910. One of the main ingredients of the
proofs is the notion of L-series in Tate algebras introduced by the third
author arXiv:1107.4511 in 2012.
