The recording, which first appeared on Facebook and has since circulated online, shows two men – one wearing a mask, and one not – arguing.
As other passengers attempt to calm the situation the men are seen scuffling and shoving one another in the aisle of the train carriage before a separate fight breaks out between two women.
Officers said they are aware of the footage on social media and that enquiries are ongoing.
Train operator TransPennine Express said the passengers' behaviour was "totally unacceptable".
In a statement, a spokesperson said: "The safety and comfort of our customers and colleagues is of paramount importance to us. The actions within the videos are totally unacceptable and have been referred to colleagues at British Transport Police."
It said passengers must follow the latest Government guidance and that face coverings must be worn on trains and at stations unless exempt.
Anyone with information or who witnessed the incident is urged to contact British Transport Police by texting 61016 or calling 0800 40 50 40 quoting reference 628 of 11/12/21.