We investigate the effect of spatial dispersion phenomenon on the performance of graphene-based plasmonic devices at terahertz (THz). For this purpose, two different components, namely a phase shifter and a low-pass filter, are taken from the literature, implemented in different graphene-based host waveguides, and analyzed as a function of the surrounding media. In the analysis, graphene conductivity is modeled first using the Kubo formalism and then employing a full-k(p) model that accurately takes into account spatial dispersion. Our study demonstrates that spatial dispersion upshifts the frequency response of the devices, limits their maximum tunable range, and degrades their frequency response. Importantly, the influence of this phenomenon significantly increases with higher permittivity values of the surrounding media, which is related to the large impact of spatial dispersion in very slow waves. These results confirm the necessity of accurately assessing nonlocal effects in the development of practical plasmonic THz devices.