What are practical uses of a protected constructor?

A protected constructor can be used to make a class effectively abstract when none of its methods are pure-virtual.

It is not quite abstract in the C++ sense since friend classes can still use it without overriding, but then you would have to declare these.

When a class is (intended as) an abstract class, a protected constructor is exactly right. In that situation you don't want objects to be instantiated from the class but only use it to inherit from.

There are other uses cases, like when a certain set of construction parameters should be limited to derived classes.

Non-public constructors are useful when there are construction requirements that cannot be guaranteed solely by the constructor. For instance, if an initialization method needs to be called right after the constructor, or if the object needs to register itself with some container/manager object, this must be done outside the constructor. By limiting access to the constructor and providing only a factory method, you can ensure that any instance a user receives will fulfill all of its guarantees. This is also commonly used to implement a Singleton, which is really just another guarantee the class makes (that there will only be a single instance).

The reason for making the constructor protected, rather than private, is the same as for making any other method or field protected instead of private: so that it can be inherited by children. Perhaps you want a public, non-virtual factory method in the base class, which returns references to instances of the derived classes; the derived classes obviously want access to the parent constructors, but you still don't want to be creating them outside of your factory.