Moderate, qualify, temper are comparable when they mean to modify something so as to avoid an extreme or to keep within due bounds.
Moderate stresses reduction of what is excessive, but it does not necessarily imply finding the happy mean.
Qualify emphasizes restriction or more precise definition that brings a thing closer to the truth or facts or that makes it less general, inclusive, or sweeping or that gives it a clearly defined quality or character of its own.
Temper strongly implies accommodation to the needs or requirements of someone or something; it need not suggest moderation or qualification, but it usually implies the addition of a counterbalancing or mitigating thing.