  The Kitaev spin liquid realizes an emergent static $\mathbb{Z}_2$ gauge field
with vison excitations coupled to Majorana fermions. We consider Kitaev models
stacked on top of each other, weakly coupled by Heisenberg interaction $\propto
J_\perp$. This inter-layer coupling breaks the integrability of the model and
makes the gauge fields dynamic. \new{Conservation laws and topology keeps
single visons immobile. However, an inter-layer vison pairs can hop with a
hopping amplitude linear in $J_\perp$ confined to the layer, but their motion
is strongly influenced by the type of stacking. For AA stacking, an interlayer
pair has a two-dimensional motion but for the AB or ABC stacking, sheet
conservation laws restrict its motion to a one-dimensional channel within the
plane. For all stackings, an intra-layer vison-pair is constrained to move
out-of-plane only.} Depending on the anisotropy of the Kitaev couplings $K_x,
K_y, K_z$, the intra-layer vison pairs can display either coherent tunnelling
or purely incoherent hopping. When a magnetic field opens a gap for Majorana
fermions, there exist two types of intra-layer vison pairs - a bosonic and a
fermionic one. Only the bosonic pair obtains a hopping rate linear in
$J_\perp$. We use our results to identify the leading instabilities of the spin
liquid phase induced by the inter-layer coupling.
