  Endowing differentiable functions from a compact manifold to a Lie group with
the pointwise group operations one obtains the so-called current groups and, as
a special case, loop groups. These are prime examples of infinite-dimensional
Lie groups modelled on locally convex spaces. In the present paper, we
generalise this construction and show that differentiable mappings on a compact
manifold (possibly with boundary) with values in a Lie groupoid form
infinite-dimensional Lie groupoids which we call current Lie groupoids. We then
study basic differential geometry and Lie theory for these Lie groupoids of
mappings. In particular, we show that certain Lie groupoid properties, like
being a proper \'etale Lie groupoid, are inherited by the current groupoid.
Furthermore, we identify the Lie algebroid of a current Lie groupoid as a
current Lie algebroid (analogous to the current Lie algebra associated to a
current Lie group).
  To establish these results, we study superposition operators given by
postcomposition with a fixed function, between manifolds of $C^\ell$-functions.
Under natural hypotheses, these operators turn out to be a submersion (an
immersion, an embedding, proper, resp., a local diffeomorphism) if so is the
underlying map. These results are new in their generality and of independent
interest.
