The Hammers drew the second leg of the contest 1-1 in London, having lost the first game 2-0 in Leverkusen.
McKinlay appeared to shove someone from the Bayer coaching staff during the contest.
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Referee Jose Maria Sanchez ran over and showed McKinlay a straight red instantly.
Leverkusen captain Jonathan Tah was also involved in a tussle with Michail Antonio.
In the end, Leverkusen remained calm and grew in the game, eventually getting an equalizer to firm up their position in the tie.