  In this article we establish fine results on the boundary behavior of
solutions to nonlocal equations in $C^{k,\gamma}$ domains which satisfy local
Neumann conditions on the boundary. Such solutions typically blow up at the
boundary like $v \asymp d^{s-1}$ and are sometimes called large solutions. In
this setup we prove optimal regularity results for the quotients $v/d^{s-1}$,
depending on the regularity of the domain and on the data of the problem. The
results of this article will be important in a forthcoming work on nonlocal
free boundary problems.
