Normally in high-level competition (like The International for Dota 2, or the LCS for League of Legends) teams will need to provide sealed peripherals to the tournament (or let them know which ones they'd like to use) and those peripherals are inventoried by the staff of the tournament itself. These peripherals are held under lock and key, and in the event of damage or disfunction, another one is unwrapped and used.
This prevents people for using cheats embedded in the peripherals themselves, and also allows for teams to have sponsor-friendly peripherals for their own use.
I've worked with esports teams in the past, and players are very particular about their settings being similar to the ones they practice on, and use at home. Things like DPI, sensitivity or the texture of their mousepads are important for them to compete at the highest level consistently.
Usually the tournament itself will have an approved list of vendors/brands they use, or it'll be worked out between the team's management and the organizers. Generally, if the mouse can be customized by a first-party solution (ie, downloading Logitech G Hub or whatever to customize Logitech peripherals) it'll be allowed.
Also, in a lot of cases each player is given their own SSD, and they're swapped in and out depending on who's playing; again, these are inventoried and stored by the tournament itself after the match is played. This allows for a consistent setting over the tournament, and also allows things to be audited if they need to be. Generally players aren't allowed phones or outside devices on the set, and that allows for security against sneaky USB keys, etc.