Singaporean Big 4's, at the junior level prioritises hiring of Singaporeans/foreigners from local universities (NTU, NUS, SMU) and also some from SIM and also Singaporeans that graduated from overseas universities. (ie Singaporeans + Singapore Uni grads). It is an unofficial policy though, and some firms have a tiny intake for foreign graduates.

KPMG and EY for example hire only locals for their associate level intakes.

So far most of the Malaysians that got into Singaporean big 4s are either Singapore local uni grads, or came here after 2+ years experience in a comparable audit firm in Malaysia. Alternatively you may want to consider joining medium firms, where they can be less selective with candidates.

Within the same firm it's difficult. For example when I wanted to move, my KL firm didn't let me, so I had to resign and apply for a different Big 4 in Singapore. KL firms don't really like Singapore firms because SG is constantly stealing KL's manpower, so most colleagues tend to move between firms during the move to SG. Example if your in EY KL, generally you might join PwC SG or KPMG SG.