I oversee the books for a non-profit using QB Desktop, and have clarified exactly how classes work in QBD.

Using Desktop Pro, you have to use Income and Expense (P&L) accounts in your transactions to affect your Class balances. If you use a Balance Sheet Account (bank acct, liability acct, etc) your transactions will not show on a Class report. This is because Classes are primarily for tracking flows of money via P&L transactions. (Apparently QB Enterprise users can run Class reports with Balance Sheet accounts, which can be useful. But non-profits cannot afford Enterprise, so that point is moot. :)

Therefore, to transfer a balance from one Class to another, my suggestion is to first create a special Income Account (or Expense Account if you prefer)--the one I made is called "4999 Fund Transfers". Then create a Journal Entry and use the special Fund Transfers account FOR BOTH THE DEBIT AND CREDIT. This way there is no change on the P&L--the income (or expense) is zero'd out. You DEBIT the Class you want to move the balance FROM, and you CREDIT the Class you want to move the money TO. Very simple. Memorize the Fund Transfer transaction so you won't have to remember which side to Debit and Credit. 

This also assumes you are not actually transferring money between bank accounts. If you have to do that as well, then Write a Check from one bank account using the Fund Transfers account (Income or Expense, doesn't matter), and then Deposit to the second bank account using that same Fund Transfers account. Again, this will zero out the P&L account, adjust the Classes as needed, and move the funds as well.