Yes, you can transfer a grade 6 student who has received KPSEA exam results. The process is much easier if you don't have any problems with the headteacher. Private schools tend to frustrate parents to avoid transfer for financial reasons.
Your headteacher should be able to assist you with the transfer process. The most important thing is the NEMIS number.
If your headteacher is not willing to help with the transfer process, then get a release and acceptance letter from the school you are leaving and joining respectively. Thereafter, visit the sub-county director of education for transfer.