when you lock the rate with a LENDER, the lender spend money hedge the rate in well street to guarantee the rate. As it cost, different lender has different float down policy. Mostly the best deal lender has no float down policy. So depend the time left for your closing date, you can discuss with the broker to select the lender with reasonable rate/closing cost and float down policy for you.
Most lenders can float down, but need rate drop by 0.25% or more. And you can not down to current best deal. EX, whenthe market drop to 3.875%, you may get 4.125. When market drop to 4.5%, you get 4.625%.
A good broker usually have many lender to select.