How does "replenish unit" work in Empire: Total War?

Replenishing units is not affected by region at all. The only restriction is if you have the cash to replenish that unit, and you may replenish as many as you want per turn.

The cost is affected by the number of troops that have been lost, where the total cost is the cost per troop for that unit multiplied by the number of lost troops. No matter where the troops are, the new recruits will show up two turns later. If some troops were lost in the meantime, the unit will still be replenished by the amount of troops that you paid for.

Replenishing units is affected by how much cash you have to pay for the "top up". It costs money to put new troops back into your depleted unit.

Also, your recruits replenish faster in cities than out in the field. Which makes sense in the scheme of things. Being in cities gives you access to resources and troops to get them back upto full strength.