After a little bit of digging, my current understanding of how it works is:
The enemy units from previous turns actually carry over to the next turn. However, at the start of each turn, a 'summoning' occurs which inserts some number of new enemy units or replaces old units with a different type. The exact number is based on a bit of randomness, turn number, player health and king health.
Each turn, some new units are added or replace old units.
- For the first three turns, only Supporter can be added
- For the next two turns, Supporter, Royalist, Snitch can be added
- For the next three turns, Guard, Royalist, Snitch can be added
- For all remaining turns or when the king life is less than 12, Archer, Knight, APC, Guard can be added.
Then, an enemy type is chosen uniformly at random from the possible options.
This enemy will then be inserted randomly at the end of an available row.
If no space can be found, the weakest enemy in the lane is replaced.
Finally, if this is the round when the king comes out, the princes and king replace the units in the first spots.
For reference,
tier 1 bones starts first turn
tier 2 starts third turn
tier 3 starts sixth turn
tier 4 starts 11th turn (or when king health drops below 12)
These do not align exactly with the stages of more difficult enemies, hence you seeing some archers and knights or snitches early.