  A pluri-Lagrangian (or Lagrangian multiform) structure is an attribute of
integrability that has mainly been studied in the context of multidimensionally
consistent lattice equations. It unifies multidimensional consistency with the
variational character of the equations. An analogous continuous structure
exists for integrable hierarchies of differential equations. We present a
continuum limit procedure for pluri-Lagrangian systems. In this procedure the
lattice parameters are interpreted as Miwa variables, describing a particular
embedding in continuous multi-time of the mesh on which the discrete system
lives. Then we seek differential equations whose solutions interpolate the
embedded discrete solutions. The continuous systems found this way are
hierarchies of differential equations. We show that this continuum limit can
also be applied to the corresponding pluri-Lagrangian structures. We apply our
method to the discrete Toda lattice and to equations H1 and Q1$_{\delta = 0}$
from the ABS list.
