Observations of solar flare ribbons and prominence eruptions suggest that the magnetic reconnection process underlying them begins locally and spreads in the out-of-plane direction as it proceeds. Much has been learned from 3D numerical simulations of quasi-2D current sheets about how reconnection spreads, including an empirical understanding of the direction and the out-of-plane spreading speed as a function of system parameters for idealized systems. It has been shown that in the absence of an out-of-plane (guide) magnetic field, the spreading occurs at the speed and direction of the current carriers; with a guide field, the spreading is bi-directional at the Alfven speed. Here, we advance upon previous knowledge in two key ways. First, we develop a first principles theory of 3D magnetic reconnection spreading. We identify the key micro- and meso-scale physics causing the spreading of reconnection with and without a guide field, and predict the scaling for the spreading speed in these configurations. The predictions reproduce the previously determined empirical results for both configurations. Second, we use the theory to predict the spreading speed for current sheets of non-uniform thickness. We confirm these predictions via a parametric study using 3D two-fluid numerical simulations. A key result is that the spreading with no guide field in non-uniform current sheets is slower than the speed of the current carriers in the thicker regions of the current sheet. The results are potentially important for understanding observations in which reconnection spreads, including why the observed spreading speed is often at sub-Alfvénic speeds. The result may also have applications to Earth’s magnetosphere, laboratory reconnection experiments, and reconnection in the solar wind.