The Aurors and the Ministry handled that.
Dumbledore wouldn’t have needed to hunt down Death Eaters himself - that was the Aurors’ job. They caught Dark wizards, which included Death Eaters. One of the Aurors at the time was Dumbledore’s friend Mad-Eye Moody, so he’d likely have trusted the Aurors to handle.
“He’s an old friend of Dumbledore’s, isn’t he?’ said Charlie.
‘Dumbledore’s not what you’d call normal, though, is he?’ said Fred. ‘I mean, I know he’s a genius and everything …’
‘Who is Mad-Eye?’ asked Harry.
‘He’s retired, used to work at the Ministry,’ said Charlie. ‘I met him once when Dad took me into work with him. He was an Auror – one of the best … a Dark-wizard-catcher,’ he added, seeing Harry’s blank look. ‘Half the cells in Azkaban are full because of him.”
- Harry Potter and the Goblet of Fire, Chapter 11 (Aboard the Hogwarts Express)
It was also the Ministry’s job to figure out who was under the Imperius Curse, and who was working for the Dark Lord of their own free will.
“Years back, there were a lot of witches and wizards being controlled by the Imperius Curse,’ said Moody, and Harry knew he was talking about the days in which Voldemort had been all-powerful. ‘Some job for the Ministry, trying to sort out who was being forced to act, and who was acting of their own free will.”
- Harry Potter and the Goblet of Fire, Chapter 14 (The Unforgivable Curses)
There wasn’t much reason for Dumbledore to get involved.
Also, Dumbledore wouldn’t want to.
Dumbledore also wouldn’t want to hunt down anyone. He considered using certain types of powers immoral.
“I know you haven’t,’ said Professor McGonagall, sounding half-exasperated, half-admiring. ‘But you’re different. Everyone knows you’re the only one You-Know – oh, all right, Voldemort – was frightened of.’
‘You flatter me,’ said Dumbledore calmly. ‘Voldemort had powers I will never have.’
‘Only because you’re too – well – noble to use them.”
- Harry Potter and the Philosopher's Stone, Chapter 1 (The Boy who Lived)
He refused a job as Minister because he didn’t trust himself with power.
“I, meanwhile, was offered the post of Minister for Magic, not once, but several times. Naturally, I refused. I had learned that I was not to be trusted with power.’
‘But you’d have been better, much better, than Fudge or Scrimgeour!’ burst out Harry.
‘Would I?’ asked Dumbledore heavily. ‘I am not so sure. I had proven, as a very young man, that power was my weakness and my temptation. It is a curious thing, Harry, but perhaps those who are best suited to power are those who have never sought it.”
- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)
Because of his temperament, it’s very unlikely that he’d choose to hunt down Death Eaters, because he didn’t quite trust himself with power.