Void type defects created during liquid composite molding processes have a major influence on the performance of composite parts used for instance in the aeronautic industry. Numerically, the void content in a part can be related to processing parameters by modelling resin flow through the composite. Resin flow is modelled in this work using a fully implicit Residual-Based Variational Multi-Scale (RBVMS) formulation. The Navier-Stokes equation is solved using stabilized finite elements with quadratic interpolation for the velocity and linear for the pressure. The interface is modelled using the level-set method with quadratic interpolation, which involves a level-set advection equation that is fully coupled to the Navier-Stokes equation using a Continuum Surface Force (CSF) model for surface tension. Consistent linearization and robustness of the fully coupled formulation with all RBVMS stabilization terms is achieved using local numerical differentiation and automatic time-step control. Added to the resin/air interface that is modelled using the level-set method, interfaces between yarns or fibres and the resin and the air must also be modelled. In this work, a conforming-nonconforming mesh generation and adaption algorithm coupled to appropriate error estimators, is used to achieve this goal. Solid phases such as yarns or fibres are meshed in a pre-processing step from real microstructure images acquired using e.g., tomography. This is done by first iteratively refining a non-conforming mesh close to interfaces using a curvature-based error estimator, and then fitting interface elements to obtain a conforming mesh of these interfaces. This conform meshing strategy is not employed for the resin/air interface due to its fast motion and complex topological changes. Instead, a dynamic adaptive strategy using an anisotropic error estimator is proposed to keep elements refined close to the interface. This strategy is inspired from Hessian-based error estimators for linear interpolation, but extended to deal with the quadratic interpolation used for the velocity and the level-set function. Simulations of resin flow through two-dimensional unidirectional fibre composites and three-dimensional woven fibre composites are proposed to illustrate the capabilities of the proposed numerical model. Air bubbles entrapment is considered with a close attention.