As Mark stated, for the most part they are the same. There may be slight differences between some of them with some bits added or subtracted as features slightly change, or they may be at different offsets in the memory map, or how they're clocked (as with Mark's example) but functionally most of the peripherals are carried over between devices. Make sure to look at Chapter 3 of the Reference Manual for the specific implementation on your selected device.
There are a few migration guides available, but not one for moving to the K0 family. But they should give you an idea of what to look for, and may be useful for anyone else that comes across this thread. There first two guides in this list have the largest number of changes since there were several improvements as we moved from the very first Kinetis devices into a more mature family:
Kinetis 100 MHz Rev 1.x to 120 MHz Migration Guide
Kinetis 100 MHz Rev 1.x to Rev 2.x Migration Guide
Kinetis KLx6/KL34 to Kinetis KL43/KL33/KL27/KL17 Migration Guide
Kinetis MK24FN1M to MK24FN256 Migration Guide