My question is if the sentence
"the process was done correctly"
is grammatically correct and in a logical sequence of words that can be understood. I seems straight forward to me but I'm not sure it would to others.
As it stands, the sentence tells us that someone or something did the process correctly, but it doesn't identify who or what that doer was. To introduce that information, you can either extend the current passive-voice construction:
The process was done [or "performed" or "followed" or "implemented"] correctly by Mr. Big.
or you can rework the sentence in active voice:
Mr. Big did [or "performed" or "followed" or "implemented"] the process correctly.