We investigate the interplay between magnetic (B) field, gravity, and turbulence in the fragmentation process of cores within the filamentary infrared dark cloud G34.43+00.24. The B field morphology across G34.43+00.24 is traced with thermal dust polarization at 350 micron with an angular resolution of 10" (0.18 pc). Local velocity gradients are derived from N_2H^+, tracing motion in the plane of sky, and are compared with the observed local B field orientations. Besides a large-scale east-west velocity gradient, we find a close alignment between local B field orientations and local velocity gradients toward the MM1/MM2 ridge. This local correlation in alignment suggesting that gas motions are influenced by the magnetic field morphology or vice versa. We analyze and quantify B field, gravity, turbulence, and their relative importance toward the MM1, MM2 and MM3 regions with various techniques over two scales, a larger clump area at pc scale and the smaller core area at 0.6 pc scale. While gravitational energy, B field, and turbulent pressure all grow systematically from large to small scale, the ratios among the three constituents develop clearly differently over scale. We propose that this varying relative importance between B field, gravity, and turbulence over scale drives and explains the different fragmentation types seen at sub-pc scale (no fragmentation in MM1; aligned fragmentation in MM2; clustered fragmentation in MM3). We discuss uncertainties, subtleties, and the robustness of our conclusion, and we stress the need of a multi-scale joint analysis to understand the dynamics in these systems.