Per the AP Stylebook, the global movement launched after the 2012 killing of Trayvon Martin with a goal to eradicate systemic racism and white supremacy and to oppose violence committed against Black people. Either Black Lives Matter as a noun or the Black Lives Matter movement is acceptable. BLM is acceptable on second reference. Although there are many groups that use “Black Lives Matter” or “BLM” in their names, only 16 are considered affiliates of the Black Lives Matter Global Network. The Black Lives Matter Global Network Foundation, which provides organizational infrastructure and funding to the affiliate chapters, was founded in 2014 after what is known as the Ferguson uprising over the August 2014 police shooting death of Michael Brown. The BLM network is acceptable on second reference.
Some respond to the Black Lives Matter movement by saying “all lives matter” or “blue lives matter,” the latter in reference to police officers. Neither is a formal movement, so lowercase and enclose in quotes.