The chromosome map diagram (ChM) proved a successful tool to identify and characterize multiple populations (MPs) in 59 Galactic globular clusters (GCs). Here, we construct ChMs for 11 GCs of both Magellanic Clouds (MCs) and with different ages to compare MPs in Galactic and extragalactic environments, and explore whether this phenomenon is universal through place and time. MPs are detected in five clusters. The fractions of 1G stars, ranging from ∼50 per cent to >80 per cent, are significantly higher than those observed in Galactic GCs with similar present-day masses. By considering both Galactic and MC clusters, the fraction of 1G stars exhibits: (i) a strong anticorrelation with the present-day mass, and (ii) with the present-day mass of 2G stars; (iii) a mild anticorrelation with 1G present-day mass. All Galactic clusters without MPs have initial masses smaller than ∼1.5 · 105 M☉ but a mass threshold governing the occurrence of MPs seems challenged by massive simple-population MC GCs; (iv) Milky Way clusters with large perigalactic distances typically host larger fractions of 1G stars, but the difference disappears when we use initial cluster masses. These facts are consistent with a scenario where the stars lost by GCs mostly belong to the 1G. By exploiting recent work based on Gaia, half of the known Type II GCs appear clustered in a distinct region of the integral of motions space, thus suggesting a common progenitor galaxy. Except for these Type II GCs, we do not find any significant difference in the MPs between clusters associated with different progenitors.