  This is the third paper in a series of work on weighted stable elliptic
surfaces - elliptic fibrations with section and marked fibers weighted between
zero and one. Motivated by Hassett's weighted pointed stable curves, we use the
log minimal model program to construct compact moduli spaces parameterizing
these objects. Moreoever, we show that the domain of weights admits a wall and
chamber structure, we describe the induced wall crossing morphisms on the
moduli spaces as the weight vector varies, and describe the surfaces that
appear on the boundary of the moduli space. The main technical result is a
proof of invariance of log plurigenera for slc elliptic surface pairs with
arbitrary weights.
