Transverse instabilities are investigated for the slab analog of the Bennett equilibrium of a magnetically self‐focused relativistic electron beam propagating through a plasma. Two kinds of plasma are considered: a channel matched to the beam and a broad uniform plasma. The high‐frequency electrostatic transverse two‐stream instability and the lower‐frequency resistive‐hose instability are treated analytically with a distributed tune (‘‘mass’’) model. Asymptotic growth is computed for each instability in various regimes, and analytic results are benchmarked against numerical solutions of the linearized system and particle‐in‐cell simulations. It is found that in the limit of large collisionless skin depth the electrostatic hose is the more virulent mode of beam breakup. Numerical examples are discussed for parameters in the range considered for plasma focusing of the asymmetric beams of interest in high‐energy physics.