I've been following this with interest.
'Learning Irish' says "a definite noun preceded by the verbal noun is in a genitive relation and therefore lenites names of people or places eg. ag moladh Cháit" ie genitive relation is not the same as the genitive form.
Other examples given: Tá sí ag pósadh fear óg - not a definite noun so no change
Bhí siad ag ól fíon agus ag imirt cluifi

Is this a regional difference maybe? I must say the possessive "a" in the original sentence is causing me confusion. This would turn 'dearthair' and 'bean' into definite nouns here, wouldn't it? And so they would be just lenited and not in the genitive form?

The actual rules on the ground are not really described in most grammar books. Historically (Classical Irish) the rules were quite simple:

1. ag + verbal noun + noun in genitive + adjective in genitive.

2. The noun was lenited if it was an indefinite or proper noun.

Today, the noun is pretty much never in the genitive if it is follows by an adjective. Also, only proper nouns are still lenited, indefinite ones are really only lenited in fixed phrases like ag baint mhóna.

Most grammar books describe a system that is a blend of Classical rules and modern Gaeltacht usage, which usually looks somewhat inconsistent and confusing.

By the way Cáit would be a proper noun and hence lenited. The phrases a fear or a bhean are definite, but not proper, so they wouldn't be lenited.

Finally, proper nouns are put in the genitive. Cáit is actually the genitive of Cáit.

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal

Thanks for that. I was forgetting the difference between definite and proper nouns. Learning Irish says genitive relation (lenition) for people and places only.
Maybe Cáit is not a good example here if it is the same in the genitive. I mean, you can't see if it's the genitive lenited or just the nominative lenited, it would still appear as Cháit (if you see what I mean).
You're right, historically the rules were a lot simpler.

Do you have the page number? (Ó Siadhail's definitely correct, he's from the area sure, I just want to line up my explanation with what he says.)

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal

It's chapter 15 page 67, in the section 'preposition ag with verbal noun'.

Thanks. Okay so, just to note, Ó Siadhail says that indefinite nouns are lenited in set phrases, e.g. ag ól bhainne, but that's about it, otherwise they wouldn't be.

However looking at rest of his explanation he says that using the verbal noun puts nouns in genitive relation. He refers back to Chapter 3 for what that means, part of the meaning being that proper nouns (placenames and people) are lenited. However other (i.e. nonproper) nouns are still in genitive relation.

Genitive relation in Irish has the following meaning for each type of noun:

1. Indefinite: Put in the genitive form. Lenite in certain fixed phrases.
2. Definite: Put in the genitive form. The definite article then lenites (masculine) or doesn't lenite (feminine) the noun.
3. Proper: Lenite and put in the genitive form.

So what Ó Siadhail is saying is that after the verbal noun, you place the next noun in genitive relation, what exactly you do being dependent on its type above.

By the way Learning Irish deals with the dialect that makes the least use of the genitive (Cois Fharraige Irish). For example, it has on p.66:
Tá Mairtín ag goil ag foghlaim Spáinnis.

In Munster for example this would have to be:
ag foghlaim Spáinnise

with Spáinnis in its genitive form. However Cois Fharraige (and Conamara Irish in general) doesn't use the genitive as much, so don't be surprised if you don't see it.

Also none of the dialects really use the genitive when there is an adjective. For instance "punching poor Seán" is much more likely to be:
ag bualadh Sheán bocht
than
ag bualadh Sheáin bhoicht.

(I think I've probably used bualadh as an example of a verbal noun about a hundred times in the past two years, I've probably punched Cáit, Seán, Tadhg, Bríd, Gobnait and everybody else at this point.)

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal

Well, I hope you don't punch me (ha!) but on page 146 he says this:
'an adjective following a masculine noun in the genitive is lenited and in the case of adjectives having one syllable the final consonant is made slender.
eg an bhaile mhóir, barr an gheata bhuí. ainm an leabhair bháin...'
However, this is in the section on the genitive and maybe it doesn't apply after the 'ag' construction?

That's something that has always struck me as well - it's often that they use that verb to explain points of grammar - it's a pity they couldn't have thought of something a little less violent.

Oh no, it does indeed. Masculine adjectives do have genitive forms and are lenited in the genitive. Just don't be surprised if you don't actually hear/read a genitive from native speakers in such situations as they often drop it when an adjective is present.

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal