Data processing agreement

Hi I sell online internationally. Those customers that do not live in the UK can send back their orders to local addresses in a few countries (so they do not need to pay internatioann shipping only local). Those local addreses are provided by a third party - and we are invoiced for that. That third party records names of the customers and what is sent back whose returns they received so we could refund these customers. Am I correct to think that in this case we should have a data processing agreement with that third party? These customers are ours, but it is this third party that is sending over the names (only sometimes the other way round when we are asking if there was no return but a specific customer when tracking says it has arrived). But I would say that they are processing those data on our behalf so they are the processor am I right?