Context: Radio-loud AGNs are thought to possess various sites of particle acceleration, which gives rise to the observed non-thermal spectra. Stochastic turbulent acceleration (STA) and diffusive shock acceleration (DSA) are commonly cited as potential sources of high-energy particles in weakly magnetized environments. Together, these acceleration processes and various radiative losses determine the emission characteristics of these extra-galactic radio sources. Aims: The purpose of this research is to investigate the dynamical interplay between the STA and DSA in the radio lobes of FR-II radio galaxies, as well as the manner in which these acceleration mechanisms, along with a variety of radiative losses, collectively shape the emission features seen in these extra-galactic sources. Methods: A phenomenologically motivated model of STA is considered and subsequently employed on a magneto-hydrodynamically simulated radio lobe through a novel hybrid Eulerian-Lagrangian framework. Results: STA gives rise to a curved particle spectrum that is morphologically different from the usual shock-accelerated spectrum. As a consequence of this structural difference in the underlying particle energy spectrum, various multi-wavelength features arise in the spectral energy distribution of the radio lobe. Additionally, we observe enhanced diffuse X-ray emission from radio lobes for cases where STA is taken into account in addition to DSA.