  We introduce and study a multi-marginal optimal partial transport problem.
Under a natural and sharp condition on the dominating marginals, we establish
uniqueness of the optimal plan. Our strategy of proof establishes and exploits
a connection with another novel problem, which we call the Monge-Kantorovich
partial barycenter problem (with quadratic cost). This latter problem has a
natural interpretation as a variant of the mines and factories description of
optimal transport. We then turn our attention to various analytic properties of
these two problems. Of particular interest, we show that monotonicity of the
active marginals can fail, a surprising difference from the two marginal case.
