APOL, 35 bytes

j(ƒ(i ¿(&(c(Ⓔ ↓(∋)) ≐(∈)) ∋ ↑(∋))))

Explanation

j(  Join a list to a string
  ƒ(  Listbuilder for
    i  Input
    ¿(  Returning if (called for each item in the input)
      &(  And (condition()
        c(  String contains
          Ⓔ  The constant string "aeiou"
          ↓(  To lowercase
            ∋  Current character
          )
        )
        ≐(  Is even
          ∈  For loop counter
        )
      )
      ∋  For loop item (executed if true)
      ↑(  To uppercase
        ∋  For loop item
      )
    )
  )
)