There are several synonyms for the phrase "goes down with" such as "sinks with", "falls with", "collapses with" and "decreases with". All these synonyms imply a decrease or decline in something. For example, if we talk about stocks, we can say that it "sinks with" the fluctuation in the market. Similarly, if we talk about someone's health, we can say that it "falls with" the change in weather or their diet. All these synonyms are commonly used in everyday conversation and writing to express the idea of a decrease or decline in something.