Does a Durable General Power of Attorney extent to voting rights?
No.
A voter who is mentally competent, but is not physically able to mark a ballot (to give a simple example, because they do not have hands or feet and aren't agile enough with their mouth to do so), may direct someone (authorized by state law to do so) in their immediate physical presence and at their contemporaneous direction to do so for them. Depending on state law, often a disclosure that it is being done is required, and often at least one witness to this being done is required.
For example, in a Colorado, the following statutory rule applies to signing the envelope of a mail-in ballots, which must be marked and then have the envelope in which they are submitted signed (and also by other provisions allows the marking of the ballot to do done in this fashion):
If the eligible elector is unable to sign, the eligible elector may
affirm by making a mark on the self-affirmation, with or without
assistance, witnessed by another person
Colo. Rev. Stat. ยง 1-7.5-107(3)(b.5)(II) (in the pertinent part) (the complete set of state statutes pertinent to elections in Colorado is found in this 553 page pdf document).
In the case of in person voting, this is usually done by an elections judge at the polling place, although some jurisdictions allow someone else, such as a family member, to assist.
Almost every U.S. state, territory, commonwealth, and district follows a similar rule, differing primarily in how the assistance rendered must be acknowledged or disclosed, and in who is legally permitted to mark a ballot on behalf of a disabled voter.
But one can't use a durable power of attorney as a basis for doing so. A durable power of attorney purports to give the agent marking the ballot discretion in how it is marked, which is not allowed.
Assistance in marking a ballot, even when a human is involved, is conceptualized more as a tool, just like glasses or a stylus or a pen, to mark the ballot, rather than as an ordinary common law agency relationship.
