  We investigate unique continuation properties and asymptotic behaviour at
boundary points for solutions to a class of elliptic equations involving the
spectral fractional Laplacian. An extension procedure leads us to study a
degenerate or singular equation on a cylinder, with a homogeneous Dirichlet
boundary condition on the lateral surface and a non homogeneous Neumann
condition on the basis. For the extended problem, by an Almgren-type
monotonicity formula and a blow-up analysis, we classify the local asymptotic
profiles at the edge where the transition between boundary conditions occurs.
Passing to traces, an analogous blow-up result and its consequent strong unique
continuation property is deduced for the nonlocal fractional equation.
