Australian supporter here of about 17 years, give or take. I've been lucky enough to get to Anfield once so far in those 17 years (Klopp's first home game!), and I hope to get there many more times in my life. If I had no other commitments and/or was 10 years younger, I'd probably move to Liverpool for a year or two so I can make those visits more frequent.
The first thing I'll say is, overseas supporters - the ones that do care enough to make the pilgrimage to Anfield at least, can have a very real and special connection with the club. Special enough that for eight months of the year, it's a part of the weekly routine to get up at/stay up till 2am-5am to watch the team play. Special enough to feel the heartbreak of a lost cup final or title race for days. That's with whatever family commitments, whatever travel commitments, whatever work commitments that might be going on, for a club that's on the other side of the world. In some ways, it's more of a madness to have that kind of dedication for and connection with a club for overseas supporters than it is for local supporters. After all, we don't live in cities that live and breathe football. Our communities are virtual (a la RAWK) or hard won by specifically seeking out other fans. Our connection with the club isn't something we inherit or are born into. It's something we very much choose.
With all of that said, there's no doubt in my mind that whatsoever that local supporters have more of a connection with the club. The fact that we are so far from the club means our connection to the club cannot be as viscerally real and lived as that of someone that's in the city. For example, as I hated Hicks and Gillett and mourned the loss of Rafa in 2010, and felt the turmoil of the Hodgson months and lead-up to FSG taking over, following every development up to that moment, I couldn't be a part of any of the collective action, in organisation or participation. For club tragedies like Hillsborough, I can be a student of the history and again, feel some measure of the anger. But those emotions will always be one or several steps removed because not being there means I don't have the ability to participate and be a part of the experience in the same way that I would if I were.
But, I do think we overseas supporters give a lot of love and devotion to the club in a way that we do to few other things in life. Outside of family and career (and my values, which Liverpool happily reflects), there's nothing else that's more important to me. In that context, I hope local supporters will begin by treating us like any other supporters - mind you (and maybe it's because I don't talk about being an overseas supporter everyday), I've never felt that I've been treated differently on RAWK.