Michael Hickmon's family filed the lawsuit in Dallas County on Tuesday ... claiming the actions of Talib and his brother, Yaqub Talib, at a Big XII Youth Sports League event ultimately led to Hickmon's tragic death on Aug. 13.

According to the suit, Hickmon was coaching the Dragons Elite Academy 9-and-under youth football team against the Talibs' North Dallas United Bobcats.

Hickmon's family says the game was contentious -- claiming several calls had tensions running "high" throughout the contest.

At one point, according to Hickmon's family, the Dragons scored a touchdown following a Bobcats' taunting penalty -- and the Talibs were so upset about it, they ended the game.

The plaintiffs claim in the suit that Hickmon then went to retrieve the game ball -- but was "confronted" by Aqib, who allegedly threw a punch.

"Surprised and outnumbered," the Hickmon family said in the suit, "Hickmon defensively backpedaled."

The family went on to say in the lawsuit that Yaqub then pulled out a handgun and fired five shots -- striking Hickmon with several bullets.

Hickmon later died at a local hospital.

The Hickmon family is suing the Talibs and others for wrongful death ... asking for unspecified damages.

As we previously reported, Yaqub was arrested and later charged with murder. 36-year-old Aqib, meanwhile, has not been charged with any crimes.

Aqib's attorney released a statement following it all in August ... saying, "Aqib was present when this unfortunate incident occurred and is very distraught and devastated over this terrible loss of life. He would like to convey his condolences to the family of the victim and to everyone who witnessed this unfortunate tragedy."