(CNN) -- The family -- mom, dad and two kids -- left their home looking like they meant to be back in a few minutes. But surveillance video caught what appears to be the four of them walking across the border into Mexico. There's been no word of them since.
Now the investigation into the disappearance of the Southern California family in 2010 is being taken over by the FBI, authorities in California announced.
Since Joseph McStay, his wife, Summer, and their two young children, Gianni, and Joseph Mateo, vanished more than three years ago, the San Diego County Sheriff's Department has been leading the search.
"Due to the fact that the family is believed to be out of the country, and because minor children are involved, the primary investigative responsibility for the case is being transferred to the FBI," Lt. Glenn Giannantonio of the San Diego County's department said in a prepared statement Tuesday.
"The FBI is the most appropriate law enforcement agency to continue the investigation since they have significant experience and investigative assets in foreign countries. The sheriff's department will continue participating in the investigation in a supporting role."
The McStays were last seen or heard from on February 4, 2010. When police entered their home in Fallbrook, California, more than 10 days later, they found eggs on the kitchen counter and bowls of popcorn in the living room. Their two dogs were left unattended in the residence.
The family's locked and abandoned Isuzu Trooper had been located on February 8 in San Ysidro, just two blocks from the Mexican border. Surveillance video taken that night shows a family fitting the description of the McStays walking into Mexico.
"We do believe the family on the tape going into Mexico is that of the McStays," Giannantonio told CNN earlier this year.
Joseph McStay's mother welcomed the FBI's involvement.
"I'm very happy for this," Susan Blake said to CNN Tuesday evening. "I'm happy to get more of the international part of that. Now, it can reach across to all other countries. It doesn't mean they are necessarily in Mexico, but it will keep their names and faces out there. It's just a big benefit."
Three years without a clue have taken a heavy toll on Blake.
Previously she had expressed doubt that the people in the video were her son and his family, noting that his asthma medicine was left in the car and there were no car seats for the children.
"I just don't know what to think anymore, "she said Tuesday, "if they are in Mexico, or even if my son is alive. These are answers that I need. And these children, I just wonder what they look like. Once they find them are they going to remember us?"
"I hope they solve it, so everybody can get some closure," said Joseph McStay's brother, Michael McStay. "It's been over three years, and it's been wearing on everybody, all the family members. So I'm hoping something comes up and we can get some resolution."
Anyone with information on the McStays is asked to call Crime Stoppers at (888) 580-8477.