An analytic technique for calculating magnetic‐moment jumps &Dgr;&mgr; of particles in magnetic traps, previously derived for particular two‐dimensional vacuum fields, is generalized to nonvacuum fields of arbitrary complexity and applied to high‐&bgr; mirror machines. The size of a jump depends on the behavior of the magnetic‐field strengthB(s) near the singularities ofBin the complexsplane, where realsmeasures position along a field line. It is demonstrated that an intrinsic complication of mirror‐machine magnetic configurations is the presence of multiple singularities ofB, which become closely spaced for field lines near the axis. An expansion inr2is used to determine &Dgr;&mgr; in the closely spaced regime. The analytic theory is compared with results from a particle‐orbit code for several axisymmetric nonvacuum fields, and is found to be in excellent agreement in both the well separated and closely spaced singularity regimes. Finite‐&bgr; effects are examined using axisymmetric model fields derived from the long, thin equilibrium approximation, with parameters representative of present (2XIIB) and future (MFTF) mirror experiments. It is shown that increasing &bgr; appreciably lowers the maximum energy at which particles are confined. The effect is particularly significant if the plasma length is small compared with the vacuum field axial scale length and if the gyroradius is small compared with the plasma radius. The degradation is substantially reduced when the gyroradius at the energy limit is comparable to the plasma radius.