Static memory is not automatically cleaned up for you. It is globally shared memory and you will still need to clean it up, unless you mean at app exit, at which point more or less everything is cleaned up.

char const* my string = "";
Is not static, use std::string instead, seriously.

Also you can just make a std::string instance and assign the value of a pointer dereferenced string to it. Complex types require you copy your data to a new variable / memory block yourself. Memory management involving anything pointer that's not a smart pointer, is completely your responsibility.

For your moving memory behind a pointer question, probably std::move documentation will give you some insights.

By statically allocated pointer you mean a smart pointer, I assume? Pointers are just that, references to a block of memory. How you deal with that memory is your responsibility.

I would go as far as recommending to stay away from raw pointers and only use std:: types like smart pointers, they deal with a lot of difficult things like memory management and copy/move/manipulation of data in safe ways.

AFIK, you cannot convert dynamic memory in static or stack memory since both need to be known at compile time. You can move it into static memory but you would still need a buffer with a constant size to be prepared at compile time. Which means it won't be a "new" static memory