@T.E.D. I couldn't any credible sources related to this either but I think the migration would've been an aftermath of something trivial(specific to one locality in the tamil region) which wasn't documented probably... hope I'll find it or somebody else will and answer it here ;) Thanks for taking your time and if you don't mind, can you comment few of those sources you dug up?
@Andrew Well what I did find IIRC was that that region and the Tamil area were all part of the same country at the time. So a people moving there would probably not have been a huge challenge if they had some reason (even whim) to want to go.
@T.E.D. Yes, they were all under the same King and I found few sources that indicated that Kerala(the region they migrated into) was forming its own culture(birth of a new language - Malayalam) at that time (kuzhalmannamagraharam.info/articles/kerala-iyer-history.html). I couldn't find any evidences and I couldn't accept this as the link I've quoted in the question says something else altogether...
@T.E.D. also when I was looking for the answers, I found a lead - a paper on "Plight of the Palakkad Iyers". Unfortunately, I'm not able to find the link/journal where it was published