Is it permissible to take a portion of a salary designated for a research project based on educational level, then give a part of it to the person in charge of the project to cover equipment expenses or other costs, and is what has been done unlawful (haram)?

If the salary is a right of the employee, then he has complete freedom to dispose of it by giving to the supervisor or not. However, if the salary is allocated for project expenses and is given to each individual according to their needs, then the permissibility of giving to the supervisor from this money depends on the permission of the granting authority. If they permit, there is no harm. Otherwise, the giver is liable for what was paid, because he has transgressed the conditions of the granting authority. Therefore, it is necessary to refer to the granting authority and inquire from them about the permissibility of giving to the supervisor from the allocated money.