The mutually-exciting Hawkes process (ME-HP) is a natural choice to model reciprocity, which is an important attribute of continuous-time edge (dyadic) data. However, existing ways of implementing the ME-HP for such data are either inflexible, as the exogenous (background) rate functions are typically constant and the endogenous (excitation) rate functions are specified parametrically, or inefficient, as inference usually relies on Markov chain Monte Carlo methods with high computational costs. To address these limitations, we discuss various approaches to model design, and develop three variants of non-parametric point processes for continuous-time edge modelling (CTEM). The resulting models are highly adaptable as they generate intensity functions through sigmoidal Gaussian processes, and so provide greater modelling flexibility than parametric forms. The models are implemented via a fast variational inference method enabled by a novel edge modelling construction. The superior performance of the proposed CTEM models is demonstrated through extensive experimental evaluations on four real-world continuous-time edge data sets.