to come near/agree
proserchomai most often describes moving up to a person or place, and can also mark drawing near to God or agreeing with teaching.
The dominant use describes people moving up to someone in order to speak, ask, touch, receive help, or act. In a smaller group of passages it marks drawing near in relation to God or Christ, while 1 Timothy 6:3 uses it for agreement with teaching.
Senses in use
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Approach a person or place · dominant
The term commonly introduces movement toward someone before a following action: in Matthew 18:21 Peter comes near before asking about forgiveness, and in Luke 10:34 the man comes near before tending the wounded man. It also frames people approaching to make requests or speak, as in Matthew 17:14, Matthew 21:23, Acts 22:27, and John 12:21; the APB renders these clauses with such wording as "approached," "came near," and "came to."
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Draw near in relation to God or Christ · common
In Hebrews 10:22, the first-person present middle subjunctive calls for drawing near with a true heart and full assurance of faith. Hebrews 11:6 speaks of the person drawing near to God, while 1 Peter 2:4 describes people as coming to the living stone; in these clauses the approach is directed toward God or Christ rather than an ordinary interpersonal encounter.
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Be in agreement with teaching · rare
In 1 Timothy 6:3, the present middle indicative occurs in the condition about someone who teaches differently and does not agree with the sound words of the Lord Jesus Christ and teaching fitting piety. The APB renders the clause "does not agree," showing a use concerning alignment with words and teaching rather than physical movement.
In the Anselm Project Bible: Across the cited verses, the APB variously uses "came near," "approached," "came to," and "draw near." In 1 Timothy 6:3 it uses "does not agree"; these are translation-level presentations rather than definitions.