Mumford introduced in the 1960ies an algebraic approach to the construction of (almost) canonical bases of sections of ample line bundles on abelian varieties that permitted him to construct quasi-projective moduli spaces. His construction was later re-interpreted by Welters as a flat projective connection before being generalized by Hitchin to the non-abelian setting. In this talk (part of joint work in progress with Michele Bolognesi, Johan Martens and Christian Pauly) I will present some facts concerning the Mumford-Welters connection in the context of an abelianization problem for certain non-abelian theta functions.