Push verb - Exert oneself continuously, vigorously, or obtrusively to gain an end or engage in a crusade for a certain cause or person; be an advocate for.
Usage example: The Dean is pushing for his favorite candidate
Promote is a synonym for push in urge topic. In some cases you can use "Promote" instead a verb "Push", when it comes to topics like speech, knowledge, assist, advertise. popular alternative
Promote verb - To provide publicity for.
Usage example: promoting a new line of toys based on the popular movie
Push is a synonym for promote in help topic. You can use "Push" instead a verb "Promote", if it concerns topics such as urge, speech, knowledge, advertise. informal substitute
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Google Ngram Viewer shows how "push" and "promote" have occurred on timeline