"Chasing down" is a common expression used to describe pursuing something or someone relentlessly. However, there are several synonyms for this phrase that can be used interchangeably. Some of these synonyms include "pursuing," "tracking," "tracing," "hunting," "shadowing," "hounding," and "trailing." Each of these terms denotes an action that requires one to actively follow or search for something or someone. These synonyms can be useful depending on the context of the situation, allowing a speaker or writer to convey the same message using slightly different language. So, whether you are chasing down a criminal or pursuing your dreams, these synonyms can help you express your intentions accurately.