  We introduce the algebra $\mathcal{G}^{\operatorname{f}}$, which is equipped
with natural maps into the algebra of (combinatorial) multiple Eisenstein
series and that of multiple q-zeta values. Similar to Racinet's approach to
formal multiple zeta values, we consider an affine scheme BM corresponding to
the algebra $\mathcal{G}^{\operatorname{f}}$. We show that Racinet's affine
scheme DM represented by the algebra of formal multiple zeta values embeds into
this affine scheme BM. This leads to a projection from the algebra
$\mathcal{G}^{\operatorname{f}}$ onto the algebra of formal multiple zeta
values. Via the above natural maps, this projection corresponds to taking the
constant terms of multiple Eisenstein series or the limit $q\to1$ of multiple
q-zeta values.
