I'd like to preface this by saying that I'm not a historian, but I have done extensive reading on the topic of Stoic philosophy and its history.

The Stoics absolutely intended for their philosophy to transcend national and socioeconomic boundaries. I'll provide some examples below, but the essence of Stoic thought on the human condition is that humans are animals blessed with divine reason, and that all humans share the quality of reason. As such, they believed that the purpose of human existence was to learn to live in accordance with their nature as semi-divine beings by practicing virtue. This extended to slaves and non-Romans, and it likely also extended to women. Musonius Rufus, for example, spoke in favor of educating women in Stoic philosophy the same as men; however, nearly every Stoic philosopher whose writings we have access to today was deeply mysoginistic. Marcus Aurelius would call extreme emotions "womanish," "weak," and "effeminate." Seneca the Younger frequently made reference to women as inherently less rational and virtuous than men. However, even in spite of their misogyny, I believe that most of the Stoics would have concurred with Rufus in saying that women ought to learn philosophy so that they can learn to be more virtuous, just as men should.

In regards to Stoic philosophy extending to non-Romans: Marcus Aurelius very clearly believed in what he called a universal "citizenship" in a global city or State, and he makes that argument multiple times throughout the meditations. For example, in Med. 4.4, he says:

we are fellow-citizens; if this is so, we are members of some political community; if this is so, the world is in a manner a state. For of what other common political community will any one say that the whole human race are members?

One of the most famous Stoics today, Epictetus, was born a slave and started practicing Stoicism as a slave. Many of the examples that he uses in regards to hardship are slaves being abused by their masters, or tyrants abusing their people. It's little wonder why he uses these examples so frequently as he famously had his leg permanently crippled by his master as a punishment. In Chapter One of the Discourses of Epictetus, he says:

I will not, for this is in my power. But I will put you in chains. Man, what are you talking about? Me, in chains? You may fetter my leg, but my will not even Zeus himself can overpower. I will throw you into prison. My poor body, you mean.

It's also worth noting that the Roman Empire began instituting laws to protect the rights of slaves after Stoic philosophy became one of the dominant Roman philosophies.

Lastly, if you're interested in learning more about how Stoic thought might have applied to women, I would recommend you read Musonius Rufus's lecture titled "By Musonius, from the Lecture Showing that Women Also Should Study Philosophy." Musonius, notably, was Epictetus's mentor, and Epictetus was one of Marcus Aurelius's primary Stoic influences.