  We define the local multiplier module of a Hilbert module in analogy to the
local multiplier algebra for $C^*$-algebras. We use properties of the local
multiplier module to lift non-closed actions on $C^*$-algebras by Hilbert
bimodules to closed actions on local multiplier algebras, and descend known
results on such closed actions down to their unclosed counterparts. We define
aperiodic dynamical inclusions and characterise them as crossed products by
inverse semigroup actions. We describe the slice structure for such inclusions
and show that all slices arise as linear combinations of slices already present
in the inverse semigroup action.
