The Toffees have however offered new contracts to long-serving captain Seamus Coleman and academy product Tom Davies.
For the second season running, Everton had to leave it late to preserve their top flight status.
The Toffees changed managers in January when Sean Dyche replaced Frank Lampard and they eventually finished in 17th position, two points above the relegation zone.
Mina joined Everton from Barcelona in the summer of 2018 but like Townsend and Begovic, his contract will expire at the end of this month and will not be renewed.
Director of football Kevin Thelwell said on the club’s official website: “Everyone at the club wishes to thank all departing players for their contribution throughout their time at Everton.
“We’re also grateful to our senior men’s players who are moving on, including Yerry – whose passion and determination for Everton was evident by how highly he was thought of by our fans – to Andros and Asmir who were consummate professionals during their two years at the club.
“We wish all of them the best with the next chapters in their careers.
“We have also offered new contracts to players and we will continue talking with them as we look to build a competitive squad for Sean Dyche and his staff for the new season.”
