Users with Account Owner, Administrator, and Bookkeeper user roles can assign banking transactions, to a specific budget in the Banking Tab.
Click the Banking Tab and select the transaction from your recent transactions list. To view older transfers, select the View All button to the right of Recent Activity.
To assign a Budget to a specific Transaction, simply click the "Not Assigned" section of a transaction's line item (under the Budget column). This will prompt a dropdown list of all of your created Budgets.
You can also click on the specific transaction and assigned a budget to it!

Employees and Budget Owners, along with Account Owners, Admins and Bookkeepers can also Assign Card Charges to a Budget or Label in the Cards Tab! Here's how!
For more information about budgets, see Viewing and Editing Budgets
To assign an expense to multiple budgets, you must split the transaction. For more information, see How do I split a transaction between multiple budgets?