Met this girl about 2 months ago. We have been on lots of dates, had great times, hold hands, kiss etc. Problem is she is going away for 3 months at the beginning of June so we discussed it and we decided there is no point in making it exclusive now but we might pick things up again when she comes back if we are both single still.

From the beginning she took a while to reply (like 2-6 hours) but recently she has been a lot longer. She still acts the same but continually tells me how busy her day has been but even then I know if I'm really busy I can reply to a text at some point during 12 or even 24 hours. She is also doing the whole changing up the kisses thing at the end of a text as well. Used to always be 2 then recently goes from 1 to 2, 2 again then 1.

I'm just beyond fed up of the whole thing now. She sent me a text yesterday and I just replied with a conversation ender because I don't want to have to deal with this now. She knows I like her, I told her in person so I don't know why she is bothering with the games. She is older than me and we are in our 20s but she is acting like a 15 year old school girl.

Why is she doing this and I'm I doing the right thing by just not falling for her games and playing it cool?

The first few times she did it I was hanging on her every text and then replying quite needy but I figured out her game, played it cool and she replied quicker but again she has gone back to the long awaited replies.

I wouldn't read too much into it, Sometimes when i'm busy I only put one kiss on the end of a text, instead of two, it depends if i'm in the middle of something.
As for the late replying, are you texting her even though she isn't replying? By this I mean, are you sending her repeated texts or anything while waiting for her to reply, sometimes this can be annoying, but if you don't do this, then maybe she really is busy, or she may just feel like she doesn't want to talk at that moment in time.