Pretty straightforward. I have lived in multiple locations (within same state) throughout 2020 and as a result find it a bit complicated to determine who I owe local taxes to? Do I prorate? Do I just go with the locality that my employer used to withhold from?
You must prorate your local income taxes based on the % of the year you spent in each municipality/taxing jurisdiction. See the FAQ for Keystone Collections, one of PA's larger local income tax collectors. See also This FAQ about moving midway through the year.
I've been in your shoes before, and I agree that the process is a bit complicated. You must file with each local income tax collector whose municipalities you resided in during the year. Some collectors handle multiple municipalities, and will help you correctly prorate between the municipalities they are in charge of. Otherwise, you will have to do so manually.
If your employer only withheld for one municipality throughout the year (for example because you didn't tell them you were moving), then you may want to contact your local income tax collector office or offices to let them know and ask them if they have any special way they wish to handle it. From personal experience, the individual local income tax collectors are sometimes able to reassign incorrectly withheld tax without you getting a refund in one municipality and having to pay in another. However, this may not be the case for your particular circumstance. Your best bet is to contact them and ask.