All I can really think of is starting with a position turbulence of 0, and then turning it up at 7 seconds. It'll affect the particles already spawned, so the whole line would fade into a wiggle, but it would be the whole line at once instead of travelling down the line.
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Just out of curiousity, have you tried retiming it using something other than a simple time stretch? Timewarp uses a different algorithm and has a few settings to play with--maybe you can dial in something that duplicates what you're seeing in FCP and DaVinci. It'd be super trial-and-error, though.