Ok. I have spemt the last hour & half getting way to intimate with remote shifters. I have 2 here, neither has any side to side play at all and both have slight freeplay forward and back. One has slightly more freeplay than the other but this was the one I last used and was very positive gear changes. The freeplay is about 10mm back and forth.
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All the freeplay seems to be coming from the actual rod setup rather than anything on the lever assembly.
Things of note are one of the springs is deformed, squashed and is 5 mm shorter than the other but seems to make no difference and was on the shifter with less freeplay. The good spring is 20mm uncompressed.
One of the nylon cups on the small ball end has slightly more wear than the other.
Changing levers over in the housings, the housing with less freeplay stayed as it was so the levers do not impact on this.
As I said the shifter/housing with slightly more freeplay was the last I used and remeber being positive. It was also a freshly rebuilt box for what that is worth. There is no small spring on either I have here and the nylon cup has a lip at the bottom so any spring would not pass through it anyway.
Hope you find this helpful or useful.