  Two-proton ($2p$) radioactivity provides fundamental knowledge on the
three-body decay mechanism and the residual nuclear interaction. In this work,
we propose decay width formulae in the extended R-matrix framework for
different decay mechanisms, including sequential $2p$ decay, diproton decay,
tri-body decay, and sequential two-diproton decay. The diproton and tri-body
formulae, combined with information on the two-nucleon transfer amplitude and
Wigner single-particle reduced width, can reproduce well experimental $2p$
radioactivity half-lives. For the case of $^{67}$Kr, theoretical predictions
for direct $2p$ decay give much larger half-lives than the recent measurement
from RIKEN. A combination of direct and sequential $2p$ emission is analyzed by
considering a small negative one-proton separation energy and a possible
enhanced contribution from the $p$-wave component. The present method predicts
that $^{71}$Sr and $^{74}$Zr may be the most promising candidates for future
study on $2p$ radioactivity. Our model gives an upper limit of 55(4) keV for
the decay width of $4p$ emission in recently found four-proton resonant
nuclide, $^{18}$Mg, which agrees with the observed width of 115(100) keV.
