Amenable uniformly recurrent subgroups and lattice embeddings

We study lattice embeddings for the class of countable groups $Gamma$ defined by the property that the largest amenable uniformly recurrent subgroup $A_Gamma$ is continuous. When $A_Gamma$ comes from an extremely proximal action and the envelope of $A_Gamma$ is co-amenable in $Gamma$, we obtain restrictions on the locally compact groups $G$ that contain a copy of $Gamma$ as a lattice, notably regarding normal subgroups of $G$, product decompositions of $G$, and more generally dense mappings from $G$ to a product of locally compact groups.