We prove a switching lemma for constant-depth circuits over the alphabet $F_p$ with generalized AND/OR gates, extending Tal's Fourier-analytic approach from the Boolean setting. The key new ingredient is a direct computation of the $L_1$ Fourier mass of AND/OR gates over $F_p$, which yields an exact closed-form expression for the expected high-degree Fourier mass after a random restriction. Combined with a Markov inequality argument, this gives a switching lemma with an explicit, prime-independent structure: $Pr[DT(f|\rho) \geq s] \leq (ep \cdot qK / ((p-1)s))^s$. As a consequence, we obtain that for any prime $p$, constant-depth circuits of sub-exponential size over $F_p$ cannot compute $1[\sum_i x_i \equiv 0 \pmod{p}]$.