Three joints, three different jobs — and only one of them is the handle
A lever handle is held on by a chain of three separate connections, and each one fails in its own way. The grub screw, a tiny set screw under the neck, presses the handle onto the square spindle — when it backs out, the lever wobbles or spins on the spindle while the backplate stays rock solid. The through-bolts pass right through the door and clamp the two backplates against the door faces — when they loosen, the whole plate lifts and you can press it back in. And the spindle itself is the square bar that carries the turning force into the lock — when it is worn round, too short, or pushed off centre, both handles can be perfectly tight and the thing still wobbles.
Which is why the diagnosis is physical rather than visual. Rock the backplate with one hand: if the plate moves against the door, the fix is outboard, in the bolts. If the plate is solid but the lever has play, the fix is inboard, in the grub screw or the spindle. That one comparison separates the two halves of this page, and it takes ten seconds.
The grub screw hides where you are not looking
It sits on the underside or the side of the handle neck, in a recess about the size of a match head. Standing in front of the door you cannot see it, which is why so many people conclude the handle has no screws at all and start taking the backplate off — turning a five-minute job into half an hour. Crouch down, or put a phone camera underneath with the flash on. The key is small: 2.5 mm or 3 mm on most European hardware, 3/32 or 1/8 inch on older American fittings. Tighten it until it is firm, not until it will not turn — these screws go into soft metal and the hex socket rounds out easily.
The spindle has to be centred, and it has to be long enough
The spindle passes through the lock and engages a square hole in each handle. For both handles to grip, it must stick out half its excess on each side — the standard way of saying it is that the spindle should be centred. A spindle that is correct in length but pushed to one side will grip one handle and rattle in the other, and no amount of tightening fixes that. A spindle that is genuinely too short does the same thing permanently. Sizes are usually 7 mm square in the UK, 8 mm for European interior doors and 10 mm for front doors, and it must be longer than the door is thick. Measure the across-flats dimension, not the diagonal, and measure the door thickness while you are there.
Wear is the other failure: the square corners round off with use. Once a corner has lost about a millimetre, it can no longer transmit torque properly — the handle turns and the latch barely moves. Spindles cost a few dollars, which makes "replace the spindle" one of the best value decisions in home repair. The opposite case is the handle's own square hole worn oval, and that one needs a new handle, not a new spindle.
And the droop is not the handle at all
If the lever hangs down after you release it, or offers no resistance on the way, the return spring has gone. That spring lives in the lock case, not in the handle — the handle's own spring, where there is one, only adds a little counter-pressure. So replacing the handle set is the expensive wrong answer, and it is a very common one. The correct fix is a new lock case, keeping the existing handles. This is the one fault on this page where the honest answer is "buy a part, not a tool".