Context. The fragmentation of filaments in molecular clouds has attracted a lot of attention recently as there seems to be a close relation between the evolution of filaments and star formation. The study of the fragmentation process has been motivated by simple analytical models. However, only a few comprehensive studies have analysed the evolution of filaments using numerical simulations where the filaments form self-consistently as part of large-scale molecular cloud evolution. Aim. We address the early evolution of parsec-scale filaments that form within individual clouds. In particular, we focus on three questions: How do the line masses of filaments evolve? How and when do the filaments fragment? How does the fragmentation relate to the line masses of the filaments? Methods. We examine three simulated molecular clouds formed in kiloparsec-scale numerical simulations performed with the FLASH adaptive mesh refinement magnetohydrodynamic code. The simulations model a self-gravitating, magnetised, stratified, supernova-driven interstellar medium, including photoelectric heating and radiative cooling. We follow the evolution of the clouds for 6 Myr from the time self-gravity starts to act. We identify filaments using the DisPerSe algorithm, and compare the results to other filament-finding algorithms. We determine the properties of the identified filaments and compare them with the predictions of analytic filament stability models. Results. The average line masses of the identified filaments, as well as the fraction of mass in filamentary structures, increases fairly continuously after the onset of self-gravity. The filaments show fragmentation starting relatively early: the first fragments appear when the line masses lie well below the critical line mass of Ostriker’s isolated hydrostatic equilibrium solution (~16 M_⊙ pc⁻¹), commonly used as a fragmentation criterion. The average line masses of filaments identified in three-dimensional volume density cubes increases far more quickly than those identified in two-dimensional column density maps. Conclusions. Our results suggest that hydrostatic or dynamic compression from the surrounding cloud has a significant impact on the early dynamical evolution of filaments. A simple model of an isolated, isothermal cylinder may not provide a good approach for fragmentation analysis. Caution must be exercised in interpreting distributions of properties of filaments identified in column density maps, especially in the case of low-mass filaments. Comparing or combining results from studies that use different filament finding techniques is strongly discouraged.