What are the origins of "ask after" and is it dying out as the way we currently speak?
Oxford Advanced Learner's dictionary says that this is a phrasal verb which implies being interested in "how somebody is, what they are doing, etc.", i.e. it's not necessarily about health or well-being, but trying to sound in a general way more personal and more caring, a step closer to the people you mention.
It seems to me that saying 'ask about' is slightly less personal that 'ask after'. Such differences must exist due to nuances in meanings.
It is more common today to ask for someone than to ask after them, but it means the same thing. It means to inquire or make inquiries regarding that person. The OED says you will still encounter ask after someone in dialect use, but it has for the most part been replaced by asking for someone.
Here are its last two citations for that sense:
- 1866 G. Macdonald Ann. Q. Neighb. xxx. (1878) 524 – To ask after their health when he met them.
- Mod. — Did any one ask for me, while I was out? When you reach that point, ask again.
Asking about someone isn’t necessarily asking for them, however.