A cash flow argument I typically see for why a convexity adjustment is necessary is the following (taken loosely from Hull 9/e, p. 143):
Say I am short an interest rate futures contract (e.g. Eurodollars). If the futures rate (the rate referenced in the contract) rises, then I am credited funds in my margin account and am able to invest these proceeds at a higher rate. Conversely, if the futures rate falls, then I have a loss in my margin account and must finance this loss but at a lower rate.
Therefore the market will set the futures rate higher since the long side of this transaction will experience the opposite effect of investing/financing the daily settlements (finance at higher rates, invest at lower rates).
What is not clear to me is that the spot rate at which I am able to invest/borrow the balance of my margin account changes in the same direction as the futures rate. The type of argument given above seems to implicitly assume the two rates always move in the same direction--is this correct? If so, does this assumption always hold?