I use a natural deduction calculus for modal propositional logic, but my question eventually is about any (sound) modal calculi with/without axioms.
Just as an example take a proof like $\square$A $\vdash$ A that any calculus based on T or stronger can prove.
Now, $\square$A $\vdash$ A looks like a false proof because in modal logic you just cannot conclude from some $\square$A to some A independent from the world they are in, but only if they play both in the same world. So are proofs like $\square$A $\vdash$ A just a short-cut for $w_{i}\square$A $\vdash$ $w_{i}$A, i.e. in your mind you always have to add $w_{i}$ before any wff?