  Background: Proton-induced nucleon knockout $(p,pN)$ reactions have been
successfully used to study the single-particle nature of stable nuclei in
normal kinematics with the distorted-wave impulse approximation (DWIA)
framework. Recently, these reactions have been applied to rare-isotope beams at
intermediate energies in inverse kinematics to study the quenching of
spectroscopic factors.
  Purpose: Our goal is to investigate the effects of various corrections and
uncertainties within the standard DWIA formalism on the $(p,pN)$ cross
sections. The consistency of the extracted reduction factors between DWIA and
other methods is also evaluated.
  Method: We analyze the $(p,2p)$ and $(p,pn)$ reactions data measured at the
R$^3$B/LAND setup at GSI for carbon, nitrogen, and oxygen isotopes in the
incident energy range of 300--450 MeV/u. Cross sections and reduction factors
are calculated by using the DWIA method. The transverse momentum distribution
of the $^{12}$C($p$,$2p$)$^{11}$B reaction is also investigated.
  Results: We have found that including the nonlocality corrections and the M\o
ller factor affects the cross sections considerably. The proton-neutron
asymmetry dependence of reduction factors extracted by the DWIA calculation is
very weak and consistent with those given by other reaction methods and
\textit{ab initio} structure calculations.
  Conclusions: The results found in this work provide a detailed investigation
of the DWIA method for $(p,pN)$ reactions at intermediate energies. They also
suggest that some higher-order effects, which is essential for an accurate
cross-section description at large recoil momentum, is missing in the current
DWIA and other reaction models.
