Being pursued is among the most frequently reported dream themes anywhere in the survey literature, and among the least supported by any interpretive tradition. That combination is worth stating at the outset, because the gap between how common the dream is and how little is known about it is exactly where invention flourishes.
Prevalence is a genuine finding. Studies of typical dreams, running from the mid twentieth century onward and repeated across different countries and populations, consistently place being chased or pursued at or near the top of the list of themes people report having dreamt at least once. It also appears frequently in nightmare studies and in the content-analysis literature. That much is documented.
Interpretation is another matter. The older traditions do not treat pursuit as a category. The Talmudic dream material in tractate Berakhot works through specific objects, animals and actions and does not address being chased as such. Artemidorus interprets encounters with animals through the animal rather than the chase, his rule that a wolf signifies a violent enemy, and a fox an enemy who works by stealth rather than openly, is the shape his readings take. The pursuer's identity is the datum; the pursuit is the setting.
So the ancient approach, where it engages at all, reads the dream as information about people in the dreamer's life. It does not read it as information about the dreamer's avoidances.