If you would like to transfer to another school in F-1 status, you will need to have your SEVIS record transferred out to that school. If you don’t request the transfer, your new school will not be able to issue a transfer I-20 for you. To do this you will need to do the following:
Failure to request a transfer within the grace period will cause you to fall out of status. After you have fallen out of status, you will have to re-enter the U.S. with an I-20 from your new school in order to study in F-1 status at that school.