A Second World War Tribal against SMS Emden

Could a Tribal-class destroyer successfully hunt down SMS Emden even though the ships come from different wars?

Yes. The later Tribal's rate of fire and modern systems would give it the advantage.

The comparison crosses two different naval eras and therefore cannot be decided by ship type alone. The later Tribal-class destroyer has the advantage through its much higher rate of fire and more modern systems. A later destroyer can be a more dangerous combatant than an earlier light cruiser despite the apparent hierarchy of class names.