The early to mid sixteenth century was economically very hard, to say the least, in England, and for the poor, the more prosperous seventeenth century was often not much better. The population expanded, doubling from 1520 to 1640, causing heavy inflation that made foodstuffs harder to access even with a rise in agricultural production resulting from the enclosure movement. So, too, was urbanization rapid, as you note in your question - particularly in London.

Part of the reason behind urbanization was the comparative opportunity in cities for wage labor for the poor. There was their own version of the "gig economy" known as the "economy of makeshifts" - essentially, people found odd jobs wherever they could.

So, to understand better how cities managed the transient economies of migrants, it's also important to understand the cultural connotations of poverty in early modern England. There are essentially two groups in the early modern English imaginary - the "deserving" and the "undeserving" poor. The deserving poor include the disabled people that you mention in your question and otherwise incapacitated people; in the seventeenth century in particular, they also included people who looked for employment but could not find it. Under the Elizabethan Poor Laws, passed in 1601, the so-called deserving poor would receive systematized relief. Parishes would collect taxes that funded relief and alms for the the "deserving" poor who could not work due to old age or disability; those capable of working who needed a little help occasionally would get a modest stipend and might even have some community members make work for them. The "undeserving" poor, on the other hand, were considered to be lazy and immoral because they were allegedly capable of working but chose not to, and often because they moved between employers (as was the dominant trend for poor workers in the English economy at the time). The "undeserving" poor were subject to campaigns of moral improvement and the criminalization of poverty; Bridewell Prison in London, for example, shipped many alleged criminals who were poor to the colonies, sometimes just for being on the street at night. Keith Wrightson brilliantly encapsulates the Poor Laws here: "So what you have in the Poor Laws is a mixture of charity and discipline. In some ways, the system was quite well devised to deal with the periodic life-cycle crises of the laboring population, though rarely generously. It was a system which in many ways demonstrated England’s relative wealth. There was a tax base there to pay for it. But it was one, of course, which also demonstrated how unequally distributed that wealth was."

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This response has liberal contributions from Wrightson's online lecture "A Polarizing Society, 1560-1640," available through Yale's website. I've also consulted Jeremy Boulton's "'The Meaner Sort': Labouring People and the Poor," in Keith Wrightson, A Social History of England, 1500-1750 (Cambridge: Cambridge University Press, 2017).