If you go on letting your dog chase cars he will be killed by being run down one day.
Is the by used in the above sentence correct? This sentence is from an error spotting exercise as per the answer key it is suggested to replace run down by run over but I don't think run down is wrong here as it also means to get killed by car although there is nuance(between run over and run down). The reason I think by is wrong here because in another sentence I read of similar construction, by wasn't used before being . The sentence was:
I am glad to hear that you narrowly escaped being run over yesterday.