Grease, lewmar's or anybody else's. just cucks up the works and with a bit of salt added makes a fine grinding paste.
Honestly, diesel is the best stuff to use for cleaning, lubricating etc. This trick was taught to me by the engineer on a whitbread maxi. You can imagine how many winches they have and what loads they bear.
After you've cleaned your winches with diesel a few times, to get rid of the grease others have over enthusiastically applied. You'll find that winch servicing takes moments to do.
If you must plug your winches up with grease never put any on the pawls. They could stick and the freewheeling handle could hurt someone.
In any enclosed item, grease is fine, but anything liable to take a dunking / get sprayed over, wet etc. out in the weather is NOT advised to be greased.
It is better to use a light oil, such as Duck OIl or as an earlier post gave ..... diesel oil. But that one is fast losing its value as Diesel gets 'dryer' and actually needs lubricity agents added to save the cylinder bores, valves etc.
So clean them out well of all old grease, literally polish all parts so that they are spotless, then assemble them oiled. You will then find that a)they turn nice and quiet without 'grease' resistance, b) are much easier to clean and service later ..... in fact a simple lift off, wipe and oil again, put it back and thats it.
Try NOT to use WD40 or any release style oils .... they actually dry the parts and make them less lubricated
Please take care to research and use the correct grease. These days there are so many Hydro-carbon and Synthetic based greases with different additives it is easy to use the wrong one and end up with a thick jelly after weathering.
I would be wary of using diesel as my lubricant. The point loadings between the bearings and drums are very high and you would be reverting to metal to metal contact ( I admit it would be clean though) There are specific greases with additives available for this kind of lubrication challenge.
As with all things mechanical, your maintenance routinue must match your type of use and conditions you sail in.
I use a 50:50 mixture of diesel and paraffin (kerosene? anyway, the stuff used for oil lamps). I don't know whether the difference is purely psychological but I do believe that this mixture seems to be better than straight diesel. However, I do smear the main spindle - very lightly - with a product called "PBC' which stands for polybutylcuprisil. This is a kind of grease that has the appearance of metallic copper due to the flakes that it contains. It is used on ships for lubricating 'slow-moving' machinery, even at elevated temperatures (eg. steam valve spindles). Even in our summer temperatures (Mediterranean - close to 40's in August!) it seems to stay put and not melt into a dirty mess like 'normal' greases.
Lewmar's own advice is to use their brand of grease.
If this was at all damaging in any way I can see no reason that they would recommend it.
Of course, Lewmar makes a bob or two on the sale of grease, but given that if you choose an inappropriate lubricant you might be facing a hefty bill for replacement parts, why bother using anything else?
Over the years (20+) I have used Duckams Keenol? which is a waterproof grease. A finger is the best method of application (after thorough cleaning) and needs very little, just enough to cover the surfaces. Pawls etc never stick. The winches are to small so under extreme load but so far are ok ......
For what its worth, Lewmar have spent a lot of time and money researching what grease gives the best results. The grease in the tubes and tubs is the same as they use - ask their service guys. The big problem and I know this from experience as I have lost count of the number of winches I have serviced - is that people put too much in. This is not a wheel bearing that needs packing. Apply it lightly with a finger and NEVER put the grease on the pawls or behind them in the pawl pockets (this goes for the pawls in the gears as well). For thepawls, use simple 3-in-1 and they will flick open nicely as long as the springs are in good order. I always used to replace the springs at every service. The other tip is that like a lot of things, little and often is best. After the initial greasing you can give them a quick once over with 3in1 as a form of interim service. I would personally avoid using any other sort of grease, especially the silicon based ones as I have seen them solidify.