The only way to avoid "paying" capital gains taxes (assuming you have capital gains) is to not sell the investments that have capital gains. The way you do this is by making a trustee to trustee in-kind transfer, which just involves transferring the actual investments from one firm to the other without selling them. This requires that the investments be something both firms offer, not something proprietary. Vanguard can help with this.
I put "paying" in quotation marks above because, depending on your cousin's income, he may be in the 0% long term capital gains tax bracket. Remember that the gains count towards the limit and are stacked on top of other income, so he may have some gains taxed at 0% and some taxed at 15% even if his other income is low enough. State taxes vary in how they handle capital gains.