I am trying to find two predicates that don't work when the positions of different types of quantifiers are switched.

The first one is when:

∀x ∃ y P( x, y ) is true but ∃ y ∀ x P( x, y ) is false

So I though when you let P(x, y) denote x < y would satisfy that one. The second predicate I need is when:

∃ x ∀ y P( x, y ) is true but ∀y ∃ x P( x, y ) is false.

A simple answer to this would be preferable like my previous one.