In Stefan Schewede's paper Formal groups and stable homotopy of commutative rings, he introduces $\Gamma$-rings (ring spectra) $DB$ for any commutative ring $B$ such that the the set of 1-dimensional commutative formal group laws over $B$ is in bijection with the set of $\Gamma$-ring maps $\mathrm{Ring}(H\mathbb{Z},DB).$ The $\Gamma$-ring $DB$ has underlying $\Gamma$-space the functor $K\mapsto \mathrm{ker}(\tilde{B}[[K]]\to\tilde{B}[[\ast]]\cong B),$ considered as a constant simplicial set.

My question is basically whether or not we can glue together such $DB$ to get some "global sections" sort of object that determines all formal group laws over all rings, and if such a thing would have some topological relationship to the Thom spectrum $MU$. Or, another perhaps similar question is, can Schwede's spectra be reinterpreted from a stacky perspective, and if so, what relationship do they bear to $\mathcal{M}_{fgl}$?

Thanks!