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Consider a random measureon a locally compact Abelian group G acting on some random element X. Mass-stationarity - introduced in (6) - means (informally) that the origin is a typical location for (X,) in the mass of . It is an intrinsic characterization of Palm versions w.r.t stationary random measures. In this paper we show that mass-stationarity w.r.t. discreteis characterized by distributional invariance under shifts of the origin by certain mass-preserving transports involving a Bernoulli randomization of the group-identity and an allocation rule. We also show that mass-stationarity w.r.t. a generalis characterized by mass-stationarity w.r.t. a Cox process driven by .

Characterization of mass-stationarity by Bernoulli and Cox transports