On a specific level, I'm thinking USA to UK but any sort of information would be fascinating. There's sometimes references to major UK bands from the early 60s picking up things like obscure blues records and being "inspired" by them and my assumption would have been that access to them would be quite difficult. Were covers of popular songs from overseas done by national artists much more popular instead?
In terms of USA to UK, there's a variety of ways that the music got heard (which is unsurprising as the 1950s and 1960s saw plenty of American music become very popular in the UK). Firstly, record labels in the UK in the 1960s were more than happy to license recordings from the US, and print the discs and distribute them using their own resources. Decca Records (one of the more prominent 1960s UK record labels) had an imprint, London American Recordings, which they used for this specific purpose. Usually, London licensed records from the US released on minor labels that didn't have international connections of their own. So if you look at the singles listed on Discogs as coming from London American Recordings, you'll see everything from Johnny Cash country recordings originally released in the US on Sun Records in the late 1950s to Al Green soul recordings released in the US on Hi Records.
Secondly, other record labels in the UK simply owned record labels in the US (or, perhaps vice versa); EMI Records in the UK purchased Capitol Records in 1955. So, for example, when EMI released 'God Only Knows' by the Beach Boys as the single on an EP in 1966 in the UK, it was releasing a record that they owned the rights to because they owned the label it was recorded for (Capitol Records, based in Hollywood in California). When it was released in the UK, EMI released it on its Capitol label (record companies often have a variety of different labels, and in EMI's case in the 1960s, the labels were each shepherded by an individual A&R person; Parlophone, the label the Beatles were on, was basically George Martin's label), which it used to point out 'this is new American music!' basically, which at times was a selling point.
As you'll know if you've bought vinyl online, importing vinyl can be relatively expensive, and so it was usually cheaper to buy American music as re-released by a British record company (e.g., the things released by London American Recordings). That said, this was dependent on the ability of the British record companies to hear the commercial potential of a recording enough to bother with it - and this didn't always happen, especially with more obscure music. So there was something of a market for imported records; record stores like NEMS in Liverpool would, at times, import stock direct from the US or elsewhere (NEMS for a while did healthy business selling imported German stock of the recordings the Beatles made in Hamburg with Tony Sheridan in 1962), while in port cities like Liverpool, where the Beatles were from, there was sometimes a trade where American sailors would bring records with them and sell them.
For the aficionados who knew that a particular record label was a sign of quality in a particular genre of music, it was possible to acquire mail-order catalogues from specialist record labels (like Chess Records, based in Chicago, which specialised in the electric blues); to get the records, you'd then post the details of which items in the catalogue you wanted along with either money or some money equivalent like a money order or bank cheque. The mail-order catalogue method is apparently how Mick Jagger acquired the rare-in-the-UK Chuck Berry record that he was holding when he met Keith Richard at Dartford train station in October 1961. Finally, some people simply went to the US and purchased music there; Van Morrison's father apparently had one of the biggest collections of black American music in Northern Ireland, and had apparently purchased most of it while located in Detroit, Michigan in the early 1950s.