In certain rings containing non-central idempotents we characterize homomorphisms, derivations, and multipliers by their actions on elements satisfying some special conditions. For example, we consider the condition that an additive map $h$ between rings $\mathcal{A}$ and $\mathcal{B}$ satisfies $h(x)h(y)h(z)=0$ whenever $x,y,z\in\mathcal{A}$ are such that $xy=yz=0$. As an application, we obtain some new results on local derivations and local multipliers. In particular, we prove that if $\mathcal{A}$ is a prime ring containing a non-trivial idempotent, then every local derivation from $\mathcal{A}$ into itself is a derivation.