This work is concerned with the formulation of hyperelastic-thermodynamic-based models for associated elastoplasticity with non-linear isotropic and kinematic hardening valid for both large elastic and large plastic deformation. On this basis, one can then introduce explicitly the assumptions of (1), small incremental plastic deformation, and (2), small elastic strain, into the general model and obtain special cases whose behaviour corresponds to that of various classical hypoelastic formulations. In particular, these are obtained on the basis of two different thermodynamic formulations for kinematic hardening with respect to the intermediate configuration. The simplest of these, in which the plastic part of the free energy does not depend explicitly on the plastic deformation, leads for example to Jaumann-or Green-Naghdi-hypoelastic-type behaviour for linear kinematic hardening in simple shear. In particular, the former case is obtained in this context when the plastic spin is assumed constant and equal to zero, and the latter case when the plastic rotation is assumed constant and equal to the identity. Allowing the plastic part of the free energy to depend explicitly on the plastic deformation yields the second thermodynamic model for kinematic hardening considered in this work. Here, again in the special case of linear hardening, Oldroyd-like behaviour for the shear stress and back stress, but not for the normal stress, is obtained in simple shear.