In my small sample of travels CX have always issued me with lounge invites, regardless of the city - Sydney, Hong Kong, Tokyo (Narita and Haneda), Mumbai and Singapore.
In my recent flight on CX162, Sydney Flounge dragon made a firm and polite request for the invite as they wanted to charge the visit back to CX. I had to dig through my bag for it.
I was a bit put off, but never thought to complain though.
Strangely enough, when flying out of Mumbai, CX gave me an F invite to the Clipper lounge, but when walking there, I saw the BA lounge (J) for the first time (blinkered in the past?) and decided to duck in.
I asked politely if I could visit and they let me in (they scribbled down my details after presenting WP card and BP).
I've done "tours" of the lounges in HK, Tokyo and Singapore (I'd call them a lounge crawl, but I was on my own) and visited other OW lounges just for the sake of visiting
However, most lounge dragons in other locations were nice - the worst have been in London and Sydney - perhaps it is a (sad?) reflection of our culture/society vs. Asian culture.