In 1926, H. Busch formulated a theorem for one single charged particle moving along a region with a longitudinal magnetic field [H. Busch, Berechnung der Bahn von Kathodenstrahlen in axial symmetrischen electromagnetischen Felde, Z. Phys. 81, 974 (1926)ZEPYAA0044-3328]. The theorem relates particle angular momentum to the amount of field lines being enclosed by the particle cyclotron motion. This paper extends the theorem to many particles forming a beam without cylindrical symmetry. A quantity being preserved is derived, which represents the sum of difference of eigenemittances, magnetic flux through the beam area, and beam rms-vorticity multiplied by the magnetic flux. Tracking simulations and analytical calculations using the generalized Courant–Snyder formalism confirm the validity of the extended theorem. The new theorem has been applied for fast modeling of experiments with electron and ion beams on transverse emittance repartitioning conducted at FERMILAB and at GSI. Thus far, developments of beam emittance manipulations with electron or ion beams have been conducted quite decoupled from each other. The extended theorem represents a common node providing a short connection between both.