The victims were killed yesterday "by armed Islamists in the village of Bih", about five kilometres (three miles) from the Burkinabe border, a provincial police official said, speaking on condition of anonymity. The official said the gunmen were travelling on motorcycle.
One of the policemen and one of the civilians died immediately as a result of the shooting, and the other two victims died of their injuries soon after, according to a Malian security source.
The FLM draws its support from the Fulani people of central Mali, and is also linked to Ansar Dine -- Arabic for "defenders of the faith" -- one of the groups that took control of Mali's vast arid north in April 2012.
According to the Malian security source, authorities were investigating whether the FLM maintained a rear base in nearby Burkina Faso.
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Mali was shaken by a coup in 2012 that cleared the way for Tuareg separatists to seize towns and cities of the north, an expanse of desert the size of Texas.
The vast desert region is now controlled by a patchwork of armed groups, loyal to the government or the Tuareg-led former rebels.