The Alabama Monument Protection Act, enacted in 2017, basically barred local governments from removing and otherwise interfering with (mainly) Confederate war memorials. Last night, an Alabama state judge struck down the statute, on the grounds that it violated the local governments' federal rights of free speech and due process. (Thanks to reader Ramer for the pointer.)
I think the judge was pretty clearly wrong, and I expect the decision to be reversed on appeal, at least as to the federal constitutional claims. Cities and counties have no federal constitutional rights against the states that created them.
The Supreme Court has made this clear in many decisions, most recently in Ysursa v. Pocatello Educ. Ass'n (2009):
"Political subdivisions of States—counties, cities, or whatever—never were and never have been considered as sovereign entities." They are instead "subordinate governmental instrumentalities created by the State to assist in the carrying out of state governmental functions." State political subdivisions are "merely ... department[s] of the State, and the State may withhold, grant or withdraw powers and privileges as it sees fit." Trenton v. N.J. (1923)....
[Any] analogy [between private corporations, which have constitutional rights against state governments, and municipal corporations] is misguided. A private corporation is subject to the government's legal authority to regulate its conduct. A political subdivision, on the other hand, is a subordinate unit of government created by the State to carry out delegated governmental functions. A private corporation enjoys constitutional protections, but a political subdivision, "created by a state for the better ordering of government, has no privileges or immunities under the federal constitution which it may invoke in opposition to the will of its creator." Williams v. Mayor of Baltimore (1933); Trenton v. N.J. (municipality, as successor to a private water company, does not enjoy against the State the same constitutional rights as the water company: "The relations existing between the State and the water company were not the same as those between the State and the City").
So if Alabama sought to require private landowners, whether individuals, nonprofit corporations, or for-profit corporations, to mantain existing monuments (Confederate or otherwise) on their property, that likely would violate the landowners' First Amendment rights. But just as the Alabama Legislature has the power -- at least from the perspective of the federal Constitution -- to restrict speech by Alabama state agencies, so it has the power to restrict speech by Alabama city and county entities, which are treated under the federal Constitution as a form of state agency.
And this is so regardless of the references in various Supreme Court decisions to the government's power to engage in "government speech," e.g., in choosing which monuments to put up (Pleasant Grove City v. Summum (2009)) or what designs to place on license plates (Walker v. Sons of Confederate Veterans (2015)). That doctrine states that the First Amendment doesn't stop governments (including local ones) from discriminating based on viewpoint in selecting their own speech. But it doesn't protect them from rules imposed by their higher-ups in the state government structure. So a state may ban cities from putting up Confederate monuments, or may ban them from removing such monuments, without violating the cities' First Amendment rights.
Now state constitutions might, if the drafters so chose, give cities or counties or both some autonomy with respect to the state legislature. Indeed, they may give branches of the state government some such autonomy. But that's a matter of state constitutional law, not of cities' or counties' First Amendment rights. (The decision doesn't seem to discuss any state constitutional claims, but I'm not sure whether some might have been brought or might still be raised on appeal.)
Finally, note that there's a much more serious, and unsettled, question whether the federal government may restrict state and local entities' rights. But that's because state governments are indeed separate sovereigns, and not just "political subdivision[s]" "created by [the federal government] for the better ordering of government."