Abstract The nature of two recently discovered radio emitters with unusually long periods of 18 minutes (GLEAM-X J1627–52) and 21 minutes (GPM J1839–10) is highly debated. Their bright radio emission resembles that of radio magnetars, but their long periodicities and lack of detection at other wavelengths challenge the neutron star (NS) interpretation. In contrast, long rotational periods are common in white dwarfs (WDs) but, although predicted, dipolar radio emission from isolated magnetic WDs has never been unambiguously observed. In this work, we investigate these long-period objects as potential isolated NS or WD dipolar radio emitters and find that both scenarios pose significant challenges to our understanding of radio emission via pair production in dipolar magnetospheres. We also perform population-synthesis simulations based on dipolar spin-down in both pictures, assuming different initial-period distributions, masses, radii, beaming fractions, and magnetic field prescriptions, to assess their impact on the ultra-long pulsar population. In the NS scenario, we do not expect a large number of ultra-long-period pulsars under any physically motivated (or even extreme) assumptions for the period evolution. On the other hand, in the WD scenario, we can easily accommodate a large population of long-period radio emitters. However, no mechanism can easily explain the production of such bright coherent radio emission in either scenarios.