(CNN) -- San Diego police and FBI officials are investigating the fatal shooting of a Mexican man by a U.S. Border Patrol agent during a rock-throwing altercation at the border fence, authorities said Wednesday.

The Tuesday evening altercation involved two Border Patrol agents and three migrants allegedly trying to cross the border from Tijuana, Mexico, authorities said.

"A thorough, multi-agency investigation is currently being conducted by the Federal Bureau of Investigation and San Diego Police Department," a Customs and Border Protection spokesman said in a statement.

Mexico's Foreign Affairs Ministry strongly condemned the shooting of the Mexican national, who authorities said fell off the border fence upon being shot and died on the Mexican side of the border.

Mexican authorities identified the man as Jose Alfredo Yanez Reyes, 40, who was shot in the left eye.

The slain man had been throwing rocks and a piece of nail-studded wood, which struck one of the Border Patrol agents in the head, the Border Patrol told police.

San Diego police were not releasing the name of the Border Patrol agent who fatally shot the migrant and said the agent has been on the force for three years.

Rock-throwing altercations at the border have been a subject of controversy between Mexican and U.S. officials.

Mexican officials have called the Border Patrol's use of gunfire in response to such rock throwing excessive.

"This past December 15, the United States and Mexico issued a joint statement about violence prevention in the border region to avoid the factors on both sides of the border that result in this type of incident. In this context, the Foreign Ministry reiterates that the use of firearms to repel attacks with stones, which is what the preliminary information indicates may have happened in this case, represents a disproportionate use of force," a ministry statement said.

Mexican authorities are also investigating the shooting, and cameras on both sides of the border may have recorded the incident, said the deputy justice of Baja California, Fermin Gomez Gomez.

In a similar incident in June 2010, a Mexican youth, Sergio Adrian Hernandez Guereca, 14, was fatally shot in the head by a U.S. Border Patrol agent who said he was being pelted by rocks while trying to arrest a migrant at the border between El Paso, Texas, and Ciudad Juarez, Mexico, authorities said.

The teen's body was found on the Mexican side of the border, and the Border Patrol agent fired his gun while on the U.S. side, authorities said.

In San Diego, police said they received an urgent radio call Tuesday evening to help Border Patrol agents who had been attacked with rocks and sticks near the fence between San Diego and Tijuana.

The incident began when a Border Patrol agent intercepted three people crossing the border fence, and the agent called for help, San Diego police said.

When a second agent arrived, the two tried to arrest the three suspects, but two of the suspects, including Yanez, fled through an opening in the fence back into Tijuana, according to San Diego and Mexican authorities.

As the third suspect began fighting with the two Border Patrol agents, Yanez allegedly climbed the border fence and threw "large rocks" and "a large piece of wood with exposed nails" at the two agents, according to San Diego police and Mexican authorities.

The wood hit the second Border Patrol agent in the head, police said. When Yanez allegedly began to throw another large rock, the first Border Patrol agent fired his service weapon and struck Yanez once, police said.

Yanez died at the scene, on the Tijuana side of the border, Mexican and San Diego authorities said.

The other suspect, a male, was arrested, but his name and age haven't been confirmed, police said.

The Border Patrol agent who was struck in the head was treated and released at a local hospital, police said.

A Mexican official in Baja California also condemned the shooting.

"It is regrettable and reprehensible that these facts are happening on our international line," said Cuauhtemoc Cardona Benavides, the secretary of government of Baja California, following a meeting with local entrepreneurs.

"There is no justification for any shots, the elimination of a life, or killing anyone in a situation of this nature (throwing rocks)," he added.