Is this a DPOA for financial matters or for medical? In the case of a DPOA for financial matters, the document itself really lays out the guidelines. When a POA is not effective immediately but rather upon your incapacity, it is called a "Springing" POA because it springs into effect only upon your incapacity. In your case, "incapacity" can be determined in one of three ways: (i) the court finds you incapacitated; (ii) a guardian/conservator is appointed; or (iii) your attorney in fact certifies that you are incapacitated. The first two approaches I'm comfortable with, however, I rarely see a DPOA that allows the attorney in fact to make the determination of incapacity. Typically, I would recommend that incapacity be determined by the certification of at least two licensed physicians. Allowing your attorney in fact to make that determination can be dangerous unless you have the highest degree of trust and confidence in your selected agent.
Assuming you are comfortable with your agent making the determination of capacity, or upon the court making a finding to that effect, or a guardian/conservator being appointed, the agent would present the pertinent documentation to the financial institution (or any other entity that the agent is authorized to deal with on your behalf), the legal department of the institution will generally review the DPOA document, and if your agent has presented them with all the proper documentation, the agent may legally being acting on your behalf to the extent permitted by the DPOA. There is no standard process or form to follow thereafter, but each institution may have their own documentation for the agent to sign but, generally, the major hurdle is satisfying the criteria in the DPOA.
Also, if your DPOA complies with California Probate Code Section 4401, then it would be considered a "Statutory Form Power of Attorney" and a third party that refuses to honor it may be obligated to pay any legal fees you incur as a result of the refusal (as long as the court finds their refusal to have been unreasonable). This definitely gives "teeth" to a POA and if you're still unsure as to how your POA works, you should discuss it with a qualified estate planning attorney who could review the document and explain it thoroughly to you.