This one's not so easy because the issue isn't where it looks like it is... "Growing" is a gerund and is properly used for a gerund: it's a verb form with the participle-like ending, "ing," and used as a noun. ") "Up" is what makes this structure funny...
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AnonymousIs it correct to write, "We do not want our children growing up in a big city"It is.
Anonymouswhat is "growing" in this case? ... is it part of an infinitive form in which the "to be" is understood?That's a possible interpretation.