The rise of contactless and wireless devices such as mobile phones and RFID chips justifies significant concerns over privacy, and calls for communication protocols that ensure some form of unlinkability. Formally specifying this property is difficult and context-dependent, and analysing it is very complex; as is common with security protocols, several incorrect unlinkability claims can be found in the literature. Formal verification is therefore desirable, but current techniques are not sufficient to directly analyse unlinkability. In [Hirschi et al., S&P'19], two conditions have been identified that imply unlinkability and can be automatically verified. This work, however, only considers a restricted class of protocols. We adapt their formal definition as well as their proof method to the common setting of RFID authentication protocols, where readers access a central database of authorised users. Moreover, we also consider protocols where readers may update their database, and tags may also carry a mutable state. We propose sufficient conditions to ensure unlinkability, find new attacks, and obtain new proofs of unlinkability using Tamarin to establish our sufficient conditions.