Should I submit a portfolio to both RISD and Brown?  Does it need to be the same portfolio?

You are not required to submit the arts supplement for the Brown Admissions application, though it is encouraged.  If you do submit your portfolio to Brown, it will be reviewed as part of your admissions material and be factored into the admissions decision.  Whereas, the BRDD supplemental essay must be the same essay submitted to both Brown and RISD, the portfolio submitted to Brown does not need to contain the same works as the portfolio submitted to RISD. 

Can students transfer into the Dual Degree Program or apply to it after they have matriculated to Brown or RISD?

It is not possible to transfer into the Brown | RISD Dual Degree Program.  However, you may consider transferring to either Brown University or to the Rhode Island School of Design.  The respective Admissions Office is the best point of contact for how to do so.  Brown and RISD students can take advantage of the cross-registration policy.  Students at Brown can take up to 4 courses at RISD and RISD students are able to take courses at Brown to fulfill degree requirements.