  We study algebraic, combinatorial and geometric aspects of weighted PBW-type
degenerations of (partial) flag varieties in type $A$. These degenerations are
labeled by degree functions lying in an explicitly defined polyhedral cone,
which can be identified with a maximal cone in the tropical flag variety.
Varying the degree function in the cone, we recover, for example, the classical
flag variety, its abelian PBW degeneration, some of its linear degenerations
and a particular toric degeneration.
