  A search for the pair production of heavy leptons as predicted by the
type-III seesaw mechanism is presented. The search uses proton-proton collision
data at a centre-of-mass energy of 13 TeV, corresponding to 139 fb$^{-1}$ of
integrated luminosity recorded by the ATLAS detector during Run 2 of the Large
Hadron Collider. The analysis focuses on final states with three or four
electrons or muons from the possible decays of new heavy leptons via
intermediate electroweak bosons. No significant deviations above the Standard
Model expectation are observed; upper and lower limits on the heavy lepton
production cross-section and masses are derived respectively. These results are
then combined for the first time with the ones already published by ATLAS using
the channel with two leptons in the final state. The observed lower limit on
the mass of the type-III seesaw heavy leptons combining two, three and four
lepton channels together is 910 GeV at the 95% confidence level.
