Teresa Morgan weighs the evidence we have for charitable activity among early Christians.
One of the changes that Constantine makes to the law is in exempting Christian churches from paying taxes (as all other individuals, most other individuals and institutions do). Part of the reasoning behind this, at least, seems to be that churches are expected to support the poor themselves and, indeed, Constantine also gives money to bishops in order for them to spend it in their churches, partly on supporting the poor. And it certainly looks from the evidence that survives as if a lot of that kind of charity is still going on, even when the church is sort of institutionalised in the fourth and fifth century.
We hear in sermons, we hear preachers exhorting people attending church to give to the poor and those in need. We hear them occasionally – less often, I have to say – commending people for doing it. (They do more of telling people to do it than commending them for doing it – but there is a little bit of commending them for doing it as well.) And also, for instance, in documents that survive on papyrus from Greek and Roman Egypt – which is a very rich source of documentary papyri in late antiquity – we find churches and also monasteries giving money to the poor, supporting the poor, employing the poor – so, helping out. So there does seem to have been a certain amount of redistribution of wealth.