Whether it is for solving nonlinear equations, optimization problems, or autonomous dynamical systems, fixed-point-type iterations are widely used in numerical sciences. However, when the goal is to iteratively solve a problem that depends on the solution of a partial differential equation, a large linear system typically needs to be solved at each iteration. Thus, accelerating fixed point schemes is a very active research theme. On-the-fly reduced-order modelling (ROM) enables the construction of a low-dimensional, self-correcting approximation of the solution to this system during the iterative process, while removing the need to do an offline training phase and any dependence to a precomputed reduced basis (e.g. a fixed geometry or mesh). This technique has been used in specific fields before, including fluid-structure interactions and topology optimization, but no general study of this method has been done to the knowledge of authors. A general method for accelerating fixed point schemes will be presented. We show that when the iteration mapping is contractive, the error of the approximate solution is guaranteed to be within the user-defined tolerance using inexact fixed-point theory. This methodology is then applied to the solution of systems of PDEs with a block Gauss-Seidel scheme. Errors due to the ROM are propagated through each iteration with respect to the computational graph of the system, which allows to estimate whether the current iteration is still within the user-defined tolerance. As a numerical example, momentum equations for an incompressible fluid coupled with heat convection and diffusion are accelerated using this method. Significant speedups are obtained while the solution stays highly accurate, which is sometimes not the case when only the residual is used as an accuracy measurement for the ROM.