Otherwise, we are stuck in a cast and right arrow loop; lethargy and right arrow; not cast and rightarrow; no lethargy, etc.
If the same character moves and takes an action, then the concentration goes down, lethargy does not undo the actions that began / began. I will even give a concrete example:
Let's use an Eldritch Knight high enough to launch Haste (and also have multiple attacks), with Haste already active. They move towards an enemy and take the attack action. Somehow, after the first attack, they drop the concentration (say Hellish Rebuke and fail their salvation). What happens to the remaining available fighter attacks? They are not taking an additional action, but simply finishing an action that has already begun.
What if you want to prepare a spell (any spell)? Would you start taking the Ready action and prepare the spell, but lethargy undoes your Ready and the next spell? Or maybe you CAN Spells ready, they would simply ignore this whole problem.
There are too many logical gaps, plummets and traps for the answer to be something else that "you can finish throwing after starting, even if it breaks the concentration."