Why might someone not show solidarity with "Black Lives Matter"?
Black Lives Matter UK has the explicit stated goal of dismantling capitalism - a goal with which many people could reasonably disagree.
Some of the founders of Black Lives Matter are "trained Marxists", so anybody who opposes Marxism/Communism could reasonably be wary of the political goals of the BLM organization.
BLM activists/protesters have been known to threaten to vandalize small businesses if they don't comply with their demands, such as a monthly donation to a local Black nonprofit, or commitments to diversity training, etc
Devout Christians might take issue with the BLM organization's stance against the Biblical definition of family, and its championing of homosexuality and transgenderism.
Black Lives Matter UK has called for prisons and detention centres to be abolished.
Nobody should be pressured into showing support for any political organization - we have the freedom to express ourselves however we like.
Anti-Semitism: Jewish people might fear the Black Lives Matter organization for their support of Palestine over Israel
BLM support looting
It's a matter of debate as to whether Black people are disproportionately brutalized by police, which is the core assumption of the Black Lives Matter movement.