As happens in any other battle you black out. lose some money and end up back at the nearest Pokemon centre. Your challenge will have to be restarted.
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You will lose some of your money (half your money in the GBA games), and be returned to the nearest Pokemon Center, where your Pokemon wil be healed.
I think she laughs at you.
Losing to Eusine results in you whiting out, as what happens when you lose a normal trainer/gym leader battle. You're returned to the last Pokemon Center and lose half of your money.
There's a Team Plasma member in the Pokemon Center at the League to teleport you to the castle.
The Poochyena flees when you are one hit away from KO