Electronically actuated systems in internal combustion engine (ICE) vehicles provide an opportunity for introducing partial autonomy in these vehicles. Controllers for total or partial autonomy of vehicle motion necessitate models of vehicular systems characterized by accurate transient and steady-state responses. We consider an ICE powertrain with a push belt type continuous variable transmission (CVT) associated with a double pinion planetary gear set. We propose a novel model for the planetary gear set as a differential-algebraic-equation (DAE) system with switching dynamics. We extend the dynamic torque converter model by incorporating the dynamics of the torque converter clutch with it. We combine CVT variator kinematics and dynamics of pulley motion and hydraulics and develop an aggregated model. Further, we construct data-driven models for the CVT’s traction coefficient and equilibrium force ratio. To operate the CVT, we design a rule-based controller that makes the CVT function at discrete steady-state ratios. Furthermore, we provide a model of a directional control valve (DCV) to capture the partial flows during the transients of the DCV. We combine these models with the existing models of powertrain components and vehicle dynamics to study the utility of the proposed models. We consider three case study examples with realistic scenarios resembling vehicle maneuver in traffic, stop-and-go motion, and reverse motion to examine the models’ ability to capture transient and steady-state characteristics and compare the resulting behaviour with the expected response.