Let V be the weighted projective variety defined by a weighted homogeneous ideal J and C a maximal cone in the Gröbner fan of J with m rays. We construct a flat family over affine m-space that assembles the Gröbner degenerations of V associated with all faces of C. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh--Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with base the toric variety associated to cone C along its universal torsor. We apply this construction to the Grassmannians of planes with their Plücker embeddings and the Grassmannian Gr(3,6) with its cluster embedding. In each case there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley--Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. This talk is based on joint work with F. Mohammadi and A. Nájera Chávez, arxiv:2007.14972.