  We generalize many recent uniqueness results on the fractional Calder\'on
problem to cover the cases of all domains with nonempty exterior. The highlight
of our work is the characterization of uniqueness and nonuniqueness of partial
data inverse problems for the fractional conductivity equation on domains that
are bounded in one direction for conductivities supported in the whole
Euclidean space and decaying to a constant background conductivity at infinity.
We generalize the uniqueness proof for the fractional Calder\'on problem by
Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full
strength of their argument. This allows us to observe that there are also
uniqueness results for many inverse problems for higher order local
perturbations of a lower order fractional Laplacian. We give concrete example
models to illustrate these curious situations and prove Poincar\'e inequalities
for the fractional Laplacians of any order on domains that are bounded in one
direction. We establish Runge approximation results in these general settings,
improve regularity assumptions also in the cases of bounded sets and prove
general exterior determination results. Counterexamples to uniqueness in the
inverse fractional conductivity problem with partial data are constructed in
another companion work.
