We investigate the behavior near zero of the integrated density of states $\ell$ for random Schrödinger operators $Φ(-Δ) + V^ω$ in $L^2(\mathbb R^d)$, $d \geq 1$, where $Φ$ is a complete Bernstein function such that for some $α\in (0,2]$, one has $ Φ(λ) \asymp λ^{α/2}$, $λ\searrow 0$, and $V^ω(x) = \sum_{ \mathbf{i}\in \mathbb{Z}^d} q_{\mathbf{i}}(ω) W(x-\mathbf{i})$ is a random nonnegative alloy-type potential with compactly supported single site potential $W$. We prove that there are constants $C, \widetilde C,D, \widetilde D>0$ such that $$ -C \leq\liminf_{λ\searrow 0} \frac{λ^{d/α}}{|\log F_q(D λ)|}{\log \ell(λ)} \qquad \text{and} \qquad \limsup_{λ\searrow 0} \frac{λ^{d/α}}{|\log F_q(\widetilde D λ)|}\log \ell(λ) \leq -\widetilde C, $$ where $F_q$ is the common cumulative distribution function of the lattice random variables $q_{\mathbf i}$. In particular, we identify how the behavior of $\ell$ at zero depends on the lattice configuration. For typical examples of $F_q$ the constants $D$ and $\widetilde D$ can be eliminated from the statement above. We combine probabilistic and analytic methods which allow to treat, in a unified manner, both local and non-local kinetic terms such as the Laplace operator, its fractional powers and the quasi-relativistic Hamiltonians.