Mass transfer between stars in binary systems profoundly impacts their evolution, yet many aspects of this process (especially the stability, mass loss, and eventual fate of such systems) remain poorly understood. One promising avenue to constrain these processes is through the identification and characterisation of systems undergoing active mass transfer. Inspired by the slow brightening preceding stellar merger transients, we worked on a method to identify Galactic mass-transferring binaries in which the donor is a Hertzsprung gap star. We constructed an initial sample of Hertzsprung gap stars using the Gaia EDR3 contribution Starhorse catalogue, and we identified candidate mass-transferring systems by selecting sources that exhibit Balmer emission features (as seen in the low-resolution Gaia XP spectra), mid-infrared excess (from WISE photometry), and photometric variability (inferred from the error in the Gaia G-band magnitude). This multi-criteria selection yielded a sample of 67 candidates, which we further analysed using complementary photometric and spectroscopic data, as well as information from cross-matched archival catalogues. Among our candidates, we identified at least nine eclipsing binaries and some sources that are potential binaries as well. Three sources in our sample are strong candidates for mass-transferring binaries with a yellow component, and three more are binaries with a Be star. Notably, three sources in our sample are strong candidates for hosting a compact companion, based on their ultraviolet or X-ray signatures. The main sources of contamination in our search are hot but highly reddened stars (primarily Oe and Be stars) as well as regular pulsating stars such as delta Scuti and Cepheid variables. As an additional outcome of this work, we present a refined new catalogue of 308 Hertzsprung gap stars, selected using improved extinction corrections and stricter emission-line criteria. This enhanced sample is expected to contain a significantly higher fraction of scientifically valuable mass-transferring binaries.