1.
If you chase someone down, you run after them or follow them quickly and catch them. (mainly AM)
Ness chased the thief down and held him until police arrived...
For thousands of years chasing down game was the main activity in which humans were involved.
PHRASAL VERB: V n P, V P n (not pron)
2.
If you chase someone or something down, you manage to find them after searching for them.
That’s when I chased her down to be the singer in my band...
Bank officials argued that it is not their job to ~ every asset of every bank debtor.
= track down
PHRASAL VERB: V n P, V P n (not pron)