Synthesis of large-area hexagonal boron nitride (h-BN) films for two-dimensional (2D) electronic applications typically requires high temperatures (~1000 oC) and catalytic metal substrates which necessitate transfer. Here, analogous to plasma-enhanced chemical vapor deposition, a non-thermal plasma is employed to create energetic and chemically-reactive states such as atomic hydrogen and convert a molecular precursor film to h-BN at temperatures as low as 500 oC directly on metal-free substrates - a process we term plasma-enhanced chemical film conversion (PECFC). Films containing ammonia borane as a precursor are prepared by a variety of solution processing methods including spray deposition, spin coating, and ink-jet printing, and reacted in a cold-wall reactor with a planar dielectric barrier discharge operated at atmospheric pressure in a background of argon or mixture of argon and hydrogen. Systematic characterization of the converted h-BN films by micro Raman spectroscopy shows that the minimum temperature for nucleation on silicon-based substrates can be lowered from 800 to 500 oC by the addition of a plasma. Furthermore, the crystalline domain size, as reflected by a decrease in the full-width-half-maximum, increased by more than 3 times (>40 cm-1 to ~13 cm-1). To demonstrate the potential of the h-BN films as a gate dielectric in 2D electronic devices, molybdenum disulfide field-effect transistors were fabricated and the field effect mobility was found to be improved by up to four times over silicon dioxide. Overall, PECFC allows h-BN films to be grown at lower temperatures and with improved crystallinity than CVD, directly on substrates suitable for electronic device fabrication.