I'm re-reading The fountain Head. You should find a copy and read the account of the initial meeting between the protagonist, Roark and the sculptor Steve Mallory and why it is that Roark helps him.
Boiled down; the reason Ayn Rand "followers" would help this lady and her children (or anyone else for that matter) is not because of pity, but because the human spirit to "live" is amongst the highest and best of values and virtue.
Now there will be those who are familiar with the character Mallory who will say "but Mallory had given up the will to live - doesn't that contradict?". True. But Mallory possessed something else of value - namely that he was perhaps one of the best sculptors in the world. Thus, Roark helped him out of his own self-interest - namely he needed the best sculpture he could get.
So how does does one tie these together? Again, Rand posits that life - and the desire to live it to it's fullest is a virtue. Also, ability - and the love of ones work is also a virtue.
Where Rand or one of here devotees might NOT lift a finger to help someone is: 1. that someone is not interested in living (either for their own enjoyment or their own purpose). 2. they are actively squandering their ability to produce and are instead seeking to "loot" both life (the will to live autonomously) and goods from those who do produce.
I'm re-reading Adam Smith's Theory of Moral Sentiments. You should find a copy...;-)