I have a good friend who did this. But, he's a member of a serious Orthodox Jewish community. In their tradition, the process starts with a matchmaker pairing you with someone. After that, you're allowed to go on 3 dates. When the third date concludes, the man is expected to propose or break-up. There are no other options.
I watched him court a few women with varied success. One apparent downside is that although they were flying to meet up, three dates just isn't enough time to really get to know somebody. There was one time that he got engaged, but broke up during the engagement process because they disagreed on so much.
As someone who's engaged, I think it's crucial to have more face-time in the courtship period. 3 dates is not enough, and said dates must be very stressful knowing that you have a deadline on the horizon.
3-6 months of dating before getting engaged seems to be the sweet spot. When you're not having sex, you get to know someone so much faster. You'd be surprised how much you can talk about when you don't have your "vagina blinders" on (h/t LDN).
The benefit to the Orthodox Jewish route is that their courtship process is much faster because their shared faith provides so much commonality. We know that having the same beliefs as your spouse is important, but you should take it a step further and find someone who shares your exact same beliefs. When you and someone share the same faith, down to the T, you instantly have so much more in common than some random girl you meet on the street.
When I say specific I mean really specific. Not just Catholic, not just trad-Cath, even more than that. I'm not sure what the Orthodox divisions are but I guess for you it'd be someone who's Armenian Orthodox. For Protestants, make sure she's the same line as you.
This is important because you're going to spend the rest of your life with this woman. For example what if you're both Catholic, but you like Latin and she likes English, there's going to be friction. Or if you don't eat meat on Fridays, and she does. Not good.
Your faith is the bedrock of your relationship, and it's crucial that you two share the same beliefs & practices.
My Jewish friend is now happily married so it worked out. But, it was not easy, and the courtship and engagement periods were challenging. These times are difficult already (speaking from experience) and they would be 100x more difficult if we couldn't spend time together face-to-face.
To sum it up, I would suggest against long-distance courtship unless she was an exemplary candidate, she's of the same faith, AND you can move to the same city before getting engaged. Other than that it's just not worth it, and you should keep looking in your local parish.