  We consider collections of $N$ chordal random curves obtained from a critical
lattice model on a planar graph, in the limit when a fine-mesh graph
approximates a simply-connected domain. We define and study candidates for such
limits in terms of conformally invariant collections of random curves,
generated via iterated Loewner equations. These curves are a natural ``domain
Markov extension'' of the earlier introduced local multiple SLE initial
segments to global multiple SLE curves. For realizing them as scaling limits,
we provide two a priori results to guarantee the precompactness of the discrete
random curves and to allow promoting a discrete domain Markov property to the
scaling limit. These results essentially only take as input certain crossing
conditions, very similar to those introduced by Kemppainen and Smirnov, and
they allow the identification of scaling limits via the martingale strategy of
classical SLE convergence proofs. The use of these results is exemplified with
convergence proofs in various lattice models.
