This is a great question, because it really helps us to think through how the Romans thought about their (early) empire, and what being part of the Roman imperium actually meant.

Firstly, let's talk about the 'free grain policy' - which the Romans would have called the annonia. It's not best thought of as a welfare policy, and there are a few good answers on here that go into what the annonia was in more detail (this one by u/XenophonTheAthenian). I don't propose to here, except to flag up two functions - to reduce the risk of famine and as a privilege of citizenship. It wasn't so much that citizens needed the annonia or they'd starve; at least much of the time, getting the annonia was a perk that reminded you of the value of being a citizen one of the perks of citizen status (edit: I've moderated the phrasing here in light of the discussion in the replies.)

Being a citizen, to the ancient Roman mind, didn't simply mean living within the Roman empire - it was a privileged legal and political status, not (or not solely) a mark of ethnicity. While all children of two properly-married citizens were themselves citizens - and accounted for the vast majority of citizens in the period we're talking about - this wasn't the only way to gain that title. Slaves freed by citizen masters could also become citizens, as they were in many ways treated as the children of their former master, as could, later on, soldiers who completed a tour of duty in the auxilia, and their descendants. There are also some indications that you could buy citizenship, but I'm not sure how true that was, or exactly how it worked. The apostle Paul is the classic case study - a self-described 'Hebrew of Hebrews', from Tarsus in Asia Minor, whose first language was Greek, who was nevertheless (at least believably) a Roman citizen by birth. However, it needs to be stressed that, under the Republic, practically all citizens live in the immediate vicinity of Rome, and that there's a clear psychological divide between the Quirites - the people of Rome who do the ruling - and the provincales who live in the territories that they rule.

The point of all that is to say that it wouldn't really make sense to have an annonia, or anything similar, for provincials - that would in many ways run contrary to the idea of what the annonia, or a citizen, or a provincial actually was. However, that doesn't mean that the Romans ignored the provincials, or were content to simply let whatever bad things happened 'out there' run their course. There was an ideology that Rome ought to act in a kind of parental capacity, supporting the provinces - in particular - against external military threats or to correct injustice, particularly when administered by Roman officials. A good case study for both is the career of Cicero, the Roman advocate and statesman who practised at the very end of the Republic between the 80s and 40s BC.

In one of his biggest early cases, Cicero took on the cause of a delegation of Sicilians, who wanted their former governor, Gaius Verres, prosecuted for a number of crimes during his tenure, all basically amounting to embezzlement, corruption and general abuse of power. In his fictional (but very well-researched) account in his novel Imperium, Robert Harris imagines Cicero having to work hard to convince anyone around him to care - after all, Sicilians are provincials and provincials don't vote. Indeed, throughout most of his speeches, which became known as the In Verrem or the Verrines, he focuses on the crimes that Verres has (allegedy) committed against Roman citizens. However, he also creates the image of the Roman people as the protectors of their provincials, and of a bond of mutual respect and fair treatment between the Roman state and its subjects - consistently calling the people of Sicily (and various other provinces that Verres is alleged to have been involved in despoiling) 'our allies and friends'. Here's a good example of how he tries to get the (Roman, citizen) jury to see itself (all translations from Perseus, abridged and adapted where necessary):

Lastly, when Gnaeus Pompey himself said in an address to the people 'that the provinces were depopulated and tyrannised over, that the courts of justice were become base and wicked, and that he desired to provide for and to remedy that evil,' the Roman people then signified their good will, not with a shout, but with a universal uproar.

In Cicero's case, these two are one and the same - he alleges that the court is corrupt so that Verres will be acquitted and the provinces will continue to be tyrannised over. The not-very-subtle message here is that ordinary Romans won't like that - that they see the role of Rome in its empire as to rule justly and fairly. It would be fair enough to ask how many of these ordinary Romans knew or cared about what went on in Sicily, but that isn't the point here - Cicero is trying to construct an ideology.

He doesn't perhaps take it to its logical conclusion in the Verrines, but there is definitely a sense that the provinces are more than conquered people to be exploited - Cicero consistently establishes a sense of a line that must not be crossed, and whips up outrage when Verres allegedly crosses that line. Here's a good example from his time governing Achaea (Greece):

I will omit all minor things; will mention nothing except what is unprecedented, nothing except what would appear incredible, if it were alleged against any other criminal. He demanded money from a Sicyonian magistrate. Do not let this be considered a crime in Verres; others have done the same. When he could not give it, he punished him; a scandalous, but still not an unheard-of act. Listen to the sort of punishment; you will ask, of what race of men you are to think him a specimen. He ordered a fire to be made of green and damp wood in a narrow place. There he left a free man, a noble in his own country, an ally and friend of the Roman people, tortured with smoke, half dead.

There's a nice rhetorical conceit going on here - it's self-evidently wrong for a Roman official to demand money from a provincial one, even wronger to try to extort it from him - but let's focus on the really terrible things Verres did. The conceit of the whole trial, of course, is that it's the Roman senate's job to put this right - by convicting Verres and, by implication, appointing governors who are more decent and more likely to uphold the norms of Roman government.

In another episode of the same speech, Cicero talks about Verres' time in Lampascus - on the eastern side of the Hellespont - and develops the idea that being part of the Roman imperium is a two-way deal:

The citizens of Lampsacus [are] men by nature, by custom, and by education most gentle; moreover, by condition, allies of the Roman people, by fortune our subjects, by inclination our suppliants.

Do not, in the name of the immortal gods, I entreat you—do not compel the allies and foreign nations to have recourse to such a refuge as [violence]; and they must of necessity have recourse to it, unless you chastise such crimes. Nothing would ever have softened the citizens of Lampsacus towards [Verres], except their believing that he would be punished at Rome.

In other words - the sort of 'aid' that a province could expect from Rome consisted of just and fair government, and redress when things went wrong and Rome's representatives abused their power. His insistent and not-entirely-accurate use of the term 'allies' shows that he wants to suggest at least a veneer of mutuality in the relationship.

Discussions of whether Romans had any welfare tend for obvious reasons to gravitate towards the annona, the grain dole, in the City of Rome (as discussed by u/UndercoverClassicist and u/XenophonTheAthenian here already).

For some information on other practices that might benefit the poor, although generally more closely resembling charity than welfare, I have an earlier answer here that discusses alimenta (financial endowments to support the upbringing of a community's children) and certain other types of public distributions and munificence that directly put food or money in peoples' hands. It's important to note that even when during the principate various emperors, in their capacity as extremely wealthy individuals, got in on setting up and funding these, they were never welfare in the sense of a broadly available social safety net paid by taxes through the government budget, but rather gifts targeted to particular communities in Italy and the provinces.

(You might recognize in alimenta an etymological link to our modern alimony.)