How do phase transitions occur in the solid phase?
Phase transitions in solids occur in the same manner as in liquids or gases, through nucleation and growth. Yes, those processes require atomic motion. Yet,atoms diffuse in solids all the time, through various mechanisms. While potentially slower, the thermodynamics of solid phase transitions are easily seen in calorimetry measurements for example. 

Nucleation and growth of precipitates is a key part of many engineering alloys. On the other hand, some phase changes (like diamond to graphite) are very hard to make happen at near-human-tolerable temperatures. 

The solid phase also sees the martensitic phase transitions, which are second-order diffusionless atomic rearrangements. The name comes from the martensite phase observed in steels. Shape memory alloys are another example of martensitic phases.

A classic materials science text would be 'Phase Transformations in Metals and Alloys", by D.A. Porter and K.E. Easterling. In my opinion, the second edition is vastly preferable to the more recent third edition where the changes mainly seemed to be reintroducing all the errors that were removed in the second edition (sigh).