Thanks.

Someone doing that sort of work... really needs to be experienced in these sorts of things. Or they might fail and cause huge issues with the structure.

Its hypothetical, I just need to know how, I'm not actually going to do it

They will all be mechanical ties. Some better than others. Some more suitable in certain situations than others.

If it is a slab, foundations are usually extended by coring into the old slab and tying rebar to the new foundation. Footing the same. If a block wall, they should remove every other corner or wall section block and connect the new wall to it. Sometimes they use bent rebar and fill the block cores.