Thermohaline mixing is one of the main processes in low-mass red giant stars that affect the transport of chemicals and, thus, the surface abundances along the evolution. The interplay of thermohaline mixing with other processes, such as downward overshooting from the convective envelope, needs to be carefully investigated. This study aims to understand the combined effects of thermohaline mixing and envelope overshooting. After implementing the thermohaline mixing process in the PARSEC stellar evolutionary code, we computed tracks and isochrones (with the TRILEGAL code) and compared them with observational data. To constrain the efficiencies of both processes, we performed detailed modelling that is suitable for globular clusters NGC 6397 and M4. Our results indicate that an envelope overshooting efficiency parameter, Λ_e = 0.6, and a thermohaline efficiency parameter, α_th = 50, are necessary to reproduce the red-giant-branch bump magnitudes and lithium abundances observed in these clusters. We find that both envelope overshooting and thermohaline mixing have a significant impact on the variation in ⁷Li abundances. Additionally, we also explore the effects of adopting solar-scaled or α-enhanced mixtures on our models. The ¹²C and the ¹²C/¹³C ratio are also effective indicators with which to probe extra mixing in red-giant-branch stars. However, their usefulness is currently limited by the lack of precise and accurate C-isotopes abundances.