  Multidimensional consistency has emerged as a key integrability property for
partial difference equations (P$\Delta$Es) defined on the "space-time" lattice.
It has led, among other major insights, to a classification of scalar
affine-linear quadrilateral P$\Delta$Es possessing this property, leading to
the so-called ABS list. Recently, a new variational principle has been proposed
that describes the multidimensional consistency in terms of discrete Lagrangian
multi-forms. This description is based on a fundamental and highly nontrivial
property of Lagrangians for those integrable lattice equations, namely the fact
that on the solutions of the corresponding P$\Delta$E the Lagrange forms are
closed, i.e. they obey a {\it closure relation}. Here we extend those results
to the continuous case: it is known that associated with the integrable
P$\Delta$Es there exist systems of PDEs, in fact differential equations with
regard to the parameters of the lattice as independent variables, which equally
possess the property of multidimensional consistency. In this paper we
establish a universal Lagrange structure for affine-linear quad-lattices
alongside a universal Lagrange multi-form structure for the corresponding
continuous PDEs, and we show that the Lagrange forms possess the closure
property.
