The eigenvalue problem of the integral neutron transport equation is studied using generalized first-flight collision probabilities. An exact transformation law for these collision probabilities describes how they change when the total cross section of the medium varies. Applying this transformation law on eigenvalue problems of the integral transport equation leads to several useful results. Thus, an explicit eigenvalue equation for the decay constant is derived, and transformed eigenvalue problems for both the multiplication factor, k, and the decay constant, α, are given in terms of the transport properties of a reference configuration, and of scaling parameters for uniform size and/or density changes. Exact scaling laws for k and α at constant mean-free-path transformations result as a special case. Finally, a general, higher order, nonlinear perturbation theory is given for both the multiplication factor and decay constant eigenvalue problems.