Magnetic reconnection at the Earth's magnetopause facilitates the transfer of mass, energy, and momentum from the solar wind into the Earth's magnetosphere. Owing to the variability of the solar wind plasma and magnetic field, the reconnection process typically involves different conditions on both inflow sides, but occasionally more symmetric conditions are encountered as well. Based on prior research we now know that the structure of the reconnection diffusion region depends substantially on the symmetry (or lack thereof) of the inflowing plasmas and magnetic fields. It is therefore of considerable interest to investigate the transition of one scenario to the other - in particular for the purpose of understanding the role of plasma mixing, heating, and of features such as pressure nongyrotropies. This presentation will involve recent theory and modeling results pertaining to these topics, and it will illuminate the means by which these kinetic processes play a role in determining the reconnection rate. Specific emphasis will be on the structure of the reconnection region, when both inflow regions are asymmetric and reconnection occurs at shear angles other than 180 degrees.