You are able to claim New Style ESA at the same time as UC. However your New Style ESA payment will be deducted £ for £ from your UC. Also your NS ESA claim will depend on the amount of your National Insurance contributions over the last 3 years. As you are claiming UC there is no such scheme as Permitted Work. Instead you have a Work Allowance. Have you tried using a Benefits calculator such as Entitledto.co.uk ?
Universal Credit and Transitional Protection are not affected by moving geographical location. What does change with moving around the country if you are a private renter are the LHA ( local housing allowance ) rates.
David
Nothing on this board constitutes legal advice - always consult a professional about specific problems