  In this paper, we introduce a general family of sequent-style calculi over
the modal language and its fragments to capture the essence of all
constructively acceptable systems. Calling these calculi \emph{constructive},
we show that any strong enough constructive sequent calculus, satisfying a mild
technical condition, feasibly admits all Visser's rules, i.e., there is a
polynomial time algorithm that reads a proof of the premise of a Visser's rule
and provides a proof for its conclusion. As a positive application, we show the
feasible admissibility of Visser's rules in several sequent calculi for
intuitionistic modal logics, including $\mathsf{CK}$, $\mathsf{IK}$ and their
extensions by the modal axioms $T$, $B$, $4$, $5$, the modal axioms of bounded
width and depth and the propositional lax logic. On the negative side, we show
that if a strong enough intuitionistic modal logic (satisfying a mild technical
condition) does not admit at least one of Visser's rules, then it cannot have a
constructive sequent calculus. Consequently, no intermediate logic other than
$\mathsf{IPC}$ has a constructive sequent calculus.
