Tie constraint when applied properly shouldn't cause any convergence issues since it's a lineae contact and It simply merges nodes together. But you should carefully select master and slave surfaces (there are some rules like in case of contact) and what is probably the most important - make sure that either there's no gap between the surfaces or the tolerance for adjustment is set properly. It shouldn't be too high because the constraint may try to merge nodes that are far away and shouldn't be merged because their elements will become highly distorted (common "zero volume" error). But if the tolerance is too low, tie constraint may not be able to merge nodes that should be merge. When you use ties always make sure that the constraints worked ans all nodes that should be merged are actually merged. There are several ways to do it - visual inspection, datacheck analysis with unconnected_regions=on, examination of warnings and eigenfrequency extraction analysis which will show the parts not tied together as they should be.
Mesh size and type doesn't have to be the same on both sides. Tie constraint is meant to be used for dissimilar meshes.