Can I kill Death Eater before Death's Touch claims its victim?

Answer: Yes you can.

Why:

After Death Eater enters the arena, the Death Touch ability goes onto the stack. In response to the Death Touch ability, champions and summons can respond with any swift ability. Thus, if a Death Eater enters the arena within a champion or summon's awareness and that champion or summon can deal the Death Eater for 4 damage at Swift speed, then the Death Eater will die before Death Touch is resolved, removing it from the stack. 

Details:

To know how to do it you'll have to understand a few details about the stack. To put something onto the stack you need to fulfill the following two criteria:

  1. All costs have to be paid
  2. All targets have to be declared and they have to be legal targets

But once it is on the stack, it does not resolve immediately allowing responses to that ability.

The important thing to note is that for an ability to be resolved from the stack two criterias need to be met:

  1. All targets needs to still legal targets
  2. The source of the ability needs to still be in the arena

Thus, if a Death Eater is killed before Death Touch is resolved the second clause for removing something from the stack is not met and the ability is removed from the stack without being resolved. 

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