Plaid does not move money via their API.
UPDATES
Dwolla now provides a white-label solution that basically does this all for you. Combine Plaid and Dwolla, and you're basically golden for payments in the US now.
Disclaimer: I co-founded a company that is a customer of Dwolla and one of their first white label customers.
Original content
Moving money with the credentials that Plaid provides requires using the Automated Clearing House (ACH) process in the US.
What won't work
Ripple Labs (currently under federal investigation), Dwolla, BitPay, etc. all require proprietary authentication with their own platforms before they will move money. You can't use them with the routing and account number that you get from Plaid. You have to adopt their entire system...or nothing.
What will work
Plaid's API provides more for you than those other APIs because you have the routing and account number. This allows you to directly enter into the ACH system yourself. All you need to move money is the senders routing and account number, the receiver's routing and account number, and the amount. Plaid gives you 2/5 of this already.
But you need to move the funds. Using an ACH processor (like Vericheck - I was a customer), you can use their API to send money to an account. Or a bank (like Silicon Valley Bank - also was a customer), where you can generate and upload a NACHA file with the instructions.
What you're in for
Compliance and banking laws are strict. Get a good lawyer to help explain what you're up against. ACH processors will want to do comprehensive background checks on you and your business. Banks will require you to deposit a portion of your proposed transactions to cover STOP payments (when a user tells their bank to cancel a payment, like voiding a check). You may have to register as a money transmitter (a small $1M in filing, registration, and legal fees for all 50 states).
Moving money is still difficult to do on behalf of a user, but if you're willing to put in the legal work, the programming is pretty simple!