We study the field dependence of the maximum current in narrow edge-type thin-film Josephson junctions with alternating critical current density. The total current I_m(H) is evaluated within nonlocal Josephson electrodynamics taking into account the stray fields that affect the difference of the order-parameter phases across the junction and therefore the tunneling currents. We find that the phase difference along the junction is proportional to the applied field, depends on the junction geometry, but is independent of the Josephson critical current density g_c, i.e., it is universal. An explicit form for this universal function is derived for small currents through junctions of the width W<<L, the Pearl length. The result is used to calculate I_m(H). It is shown that the maxima of I_m(H) and the zeros of I_m(H) are equidistant but only in high fields. We find that the spacing between zeros is proportional to 1/W². The general approach is applied to calculate I_m(H) for a superconducting quantum interference device with two narrow edge-type junctions. If g_c changes sign periodically or randomly, as it does in grain boundaries of high-T_c materials and superconductor-ferromagnet-superconductor heterostructures, I_m(H) not only acquires the major side peaks, but due to nonlocality the following peaks decay much slower than in bulk junctions.