  In this paper we detail the thermodynamics of two flavor nonlocal
Polyakov-Nambu-Jona-Lasinio models for different parametrizations of the quark
interaction regulators. The structure of the model is upgraded in order to
allow for terms in the quark selfenergy which violate Lorentz invariance due to
the presence of the medium. We examine the critical properties, the phase
diagram as well as the equation of state. Furthermore, some aspects of the Mott
effect for pions and sigma mesons are discussed explicitly within a nonlocal
Polyakov-Nambu-Jona-Lasinio model. In particular, we continued the meson
polarization function in the complex energy plane and under certain
approximations, we were able to extract the imaginary part as a function of the
meson energy. We were not able to calculate the dynamical meson mass, and
therefore resorted to a technical study of the temperature dependence of the
meson width by replacing the meson energy with the temperature dependent
spatial meson mass. Our results show that while the temperature behavior of the
meson widths is qualitatively the same for a wide class of covariant
regulators, the special case where the nonlocal interactions are introduced via
the instanton liquid model singles out with a drastically different behavior.
