  An inclusive search for nonresonant signatures of beyond the standard model
(SM) phenomena in events with three or more charged leptons, including
hadronically decaying $\tau$ leptons, is presented. The analysis is based on a
data sample corresponding to an integrated luminosity of 138 fb$^{-1}$ of
proton-proton collisions at $\sqrt{s} =$ 13 TeV, collected by the CMS
experiment at the LHC in 2016-2018. Events are categorized based on the lepton
and b-tagged jet multiplicities and various kinematic variables. Three
scenarios of physics beyond the SM are probed, and signal-specific boosted
decision trees are used for enhancing sensitivity. No significant deviations
from the background expectations are observed. Lower limits are set at 95%
confidence level on the mass of type-III seesaw heavy fermions in the range
845-1065 GeV for various decay branching fraction combinations to SM leptons.
Doublet and singlet vector-like $\tau$ lepton extensions of the SM are excluded
for masses below 1045 GeV and in the mass range 125-150 GeV, respectively.
Scalar leptoquarks decaying exclusively to a top quark and a lepton are
excluded below 1.12-1.42 TeV, depending on the lepton flavor. For the type-III
seesaw as well as the vector-like doublet model, these constraints are the most
stringent to date. For the vector-like singlet model, these are the first
constraints from the LHC experiments. Detailed results are also presented to
facilitate alternative theoretical interpretations.
