Checking is not an expense account; it is part of Banking = a Source/Destination of funds. You use these funds on things like Expenses.
"We track our restricted donations and awards using a restricted net
asset account."
You first acknowledge them as Income. That way, they already update Equity for you. If you want to Rebalance Equity between restricted and unrestricted, then along with (for instance) a $100 donation as income, you manually and additionally Debit Unrestricted Net Asset and Credit Restricted Equity (increases it), for the Fund balance there to increase.
"This allows easy monitoring of the scholarship fund on
our statement of financial position report."
Yes.
"I now want to move funds
from our checking account to the equity account."
See, that's the mistake you are making. Checking is part of Asset and already is contributing to Equity. You don't do what you are asking.
"What is the best
mechanism for doing this"
What you do, for instance, is Spend restricted funds as Expense. That would mean, additionally, another Manual entry that will Debit Restricted Equity (decreases it) to show you Used funds as promised, relieving the restriction.
You simply track income and expense as usual. You additionally Rebalance equity for each activity, so that the Current status of Restricted Funds is clear.
If you use Class Tracking for Restricted and Unrestricted, that helps, for the Income statement, as well. That way, your Net Income seen in Equity should always be part of Unrestricted Net Assets in Equity, since that is where they are "closed to" are fiscal year end.
