Let \(X_0\) be the special fiber of a Mumford degeneration \(\pi:X\to \mathbb A^1\). Let \(X_0 = Y_1 \cup ... \cup Y_n\) be the decomposition of \(X_0\) into its irreducible components. Each \(Y_i\) is a \(d\)-dimensional toric variety by virtue of being a prime divisor of \(X\), and hence is equipped with an action of \(\mathbb G_m^d\). These actions are compatible.
More concretely, recall that \(X\) can be obtained by applying the polyhedra construction to the upper convex hull \(\tilde\sigma\) of a piecewise affine function \(\varphi:\sigma\to \mathbb R\). The affine patches of \(X\) are given by rings of the form \[\mathbb C[C_v \cap (M\oplus \mathbb Z)]\] where \(C_v\) is the cone in \(M_\mathbb R\times \mathbb R\) obtained by translating \(\tilde\sigma\) so that \(v\) is at the origin and then taking the \(\mathbb R_{\geq 0}\) span. Each cone \(C_v\) contains \(\{0\}\times \mathbb R_{\geq 0}\), so these rings are actually \(\mathbb C[t]\)-algebras given by \(t\mapsto z^{(0,1)}\). The element \(z^{(0,1)}\) is homogeneous with respect to the \(M\oplus \mathbb Z\) grading on \(\mathbb C[C_v \cap (M\oplus \mathbb Z)]\) (it has \(M\)-degree \(0\)) and so \(\mathbb C[C_v \cap (M\oplus \mathbb Z)]/(z^{(0,1)})\) inherits the \(M\)-grading. These rings correspond to the affine cover of \(X_0\), so we get a \(\mathbb G_m^d = \Spec \mathbb C[M]\) action on \(X_0\).