This study investigates convective transport phenomena of combined electro-osmotic and pressure-driven flow in a microchannel subject to constant surface heat flux, with Joule heating effect taken into account. The governing system of equations includes the electric potential field, flow field, and energy equations. Analytical solutions are obtained for constant fluid properties, while numerical solutions are presented for variable fluid properties. For constant properties, the problem is found to be governed by three ratios: the length scale ratio (the ratio of Debye length to half channel height), the velocity scale ratio (the ratio of pressure-driven velocity to electro-osmotic velocity), and the ratio of Joule heating to surface heat flux. A small length scale ratio corresponds to a microchannel, while finite length scale ratio represents a nanochannel. For electro-osmotic flow only, the momentum transport is solely a function of the length scale ratio. For combined electro-osmotic and pressure-driven flow, the velocity profile and therefore the friction factor depend on both the length scale ratio and the velocity scale ratio. Assuming a thermally fully developed flow, analytical expressions for the normalized temperature profile and Nusselt number are developed. The representative results for the friction factor, normalized temperature profile, and Nusselt number are illustrated for some typical values of the three ratios. For purely electro-osmotic flow, it is found that the Nusselt number increases with decreasing $\epsilon $, approaching the value for slug flow as the length scale ratio approaches zero. For mixed flow with a given length scale ratio, the results show that the Nusselt number decreases with the velocity scale ratio, approaching the classical Poiseuille flow as the velocity scale ratio approaches infinite. When the effects of variable fluid properties are included in the analysis, numerical solutions are generated to explore the influence of thermal conductivity and viscosity variations with local temperature on the hydrodynamic and thermal characteristics of the fluid. These temperature-dependent property variations would initially develop pressure-driven flow, and correspondingly the dimensionless velocity and volume flow rate increase to account for such variations. The friction factor reduces considerably with viscosity variation, while the Nusselt number increases gently. Although the influence of thermal conductivity variation on the hydrodynamic characteristics is not impressive, it has certain impact on the heat transfer results; more specifically, increasing the conductivity variation will produce a sensible increase in Nusselt number but a small decrease in the normalized temperature.