The things that caused these young men to fall:
Eutychus—carelessness. We should go after such, as Paul did, for they have not gone out under any charge from the assembly. They are usually just cold and/or discouraged.
Ahaziah—rebellion. We should not go after such because they are on a definite course of sin. If we do, we are liable to get the fire of God’s judgment come down upon us (governmentally), as was the case of the king’s soldiers.
Mephibosheth—innocently led away by division makers. We should seek to reach them, as the opportunity presents itself (as David did) because they have not been actively involved in dividing God’s people, but rather, they are victims of others’ mistakes.