The NN-πNN equations that describe, in a unified framework, pion production in nucleon-nucleon scattering, and pion-deuteron and nucleon-nucleon elastic scattering, have been extended to include the N(939) and Δ(1232) on an equal footing. This extension, motivated by the quark models of hadrons, has the bare N and Δ as three quark states with the same spacial wave function, but different spin isospin states. The final equations, referred to as the BB-πBB equations, are consistent with the chiral bag models to the extent that the πNN, πNΔ, and πΔΔ coupling constants and form factors are related, and can be taken from bag models. The resultant equations satisfy two- and three-body unitarity, and are derived by exposing the lowest unitarity cuts in the n-body Greens function. These equations retain important contributions missing from the NN-πNN equations. For pion production and N-N scattering they include the contribution of backward pions in the NN→NΔ transition potential, which may overcome the problem of small pp d cross section as predicted by the NN-πNN equations. For π-d elastic scattering they include an additional NN→ΝΔ tensor force that can influence the tensor polarization.