Aha! This collection of words is linked to...
Pokémon! Specifically, we are looking at the 9 different 'generations' of Pokémon (i.e. the 9 different batches of new Pokémon that have been released since the game's debut). Then within those generations we are looking for the Pokémon that have three evolutionary stages (hence the title) - this means sets of 3 Pokémon where the first evolves into the second, which then evolves into the third. Specifically, we are looking for such three-stage evolutionary lines where the names of the three Pokémon each contain as a substring a 'common' word, and these words are presented in a single row of the table.
For example, the top row of Group 1 is 'cat / met / but' - and these are...
...substrings found within the names of Caterpie, Metapod, and Butterfree, where Caterpie evolves into Metapod (at Level 7), and then evolves into Butterfree (at Level 10).
A few notes about the rules behind the OP's choices:
1. Note that these strings do not necessarily need to appear at the beginnings of these names - the second row (bat / bat / rob) represents Zubat, Golbat and Crobat, where the substrings of interest appear at the end or somewhere in the middle of the words.
2. This same example shows that in cases where two or more 'common words' appear as substrings in a name, the one that is 'more common' is presented. For 'Crobat', 'bat' is the 1676th most common word in Google's Trillion Word Corpus but 'rob' is the 1053rd and so is selected as being more common.
3. Note that this is not a list of all three-stage evolutionary lines in Pokémon - only those where every Pokémon in the chain has a three-letter 'common word' as a substring in its name (e.g. so not Charmander, Charmeleon and Charizard, even though 'Charmander' contains 'man' and 'Charmeleon' contains 'arm' - 'Charizard' contains no common three-letter word).
4. Where the later evolutions of a Pokémon were not released until a later generation, these Pokémon lines are still eligible for inclusion, and listed in the table corresponding to the earliest generation in which one of them appeared - for instance, 'man / rim / lap' represents Mankey, Primeape and Annihilape and is listed in Group 1 even though Annihilape was only introduced to the game in Generation IX (the most recent one).
This all being explained, here is a complete list of these tables' intended contents:
Group 1:
Caterpie / Metapod / Butterfree
Zubat / Golbat / Crobat
Mankey / Primeape / Annihilape
Machop / Machoke / Machamp
Dratini / Dragonair / Dragonite
Group 2:
Azurill / Marill / Azumarill
Swinub / Piloswine / Mamoswine
Larvitar / Pupitar / Tyranitar
Group 3:
Mudkip / Marshtomp / Swampert
Zigzagoon / Linoone / Obstagoon
Trapinch / Vibrava / Flygon
Bagon / Shelgon / Salamence
Group 4:
Starly / Staravia / Staraptor
Group 5:
Lillipup / Herdier / Stoutland
Roggenrola / Boldore / Gigalith
Gothita / Gothorita / Gothitelle
Vanillite / Vanillish / Vanilluxe
Litwick / Lampent / Chandelure
Group 6: No eligible trios
Group 7:
Rowlet / Dartrix / Decidueye
Popplio / Brionne / Primarina
Group 8:
Grookey / Thwackey / Rillaboom
Blipbug / Dottler / Orbeetle
Hatenna / Hattrem / Hatterene
Group 9:
Tinkatink / Tinkatuff / Tinkaton
My breakthrough?
I considered that these might all be substrings of specific sets of words at quite an early stage. I tried to think of things that came in 9 discrete sets but nothing immediately came to mind - groups of 9 aren't all that common - and I eventually realised that many of these featured in the names of countries and capitals. While this eventually turned out to be a false conclusion, the reasoning involved proved helpful in finding the true solution path...
One word that cropped up several times was 'ago'. I had this pegged as a substring of 'Trinidad and Tobago' - but this country is a small island nation, with little reputation in the worlds of geography, sports, or science (three common areas of pub quiz knowledge), so it was unlikely for this nation to feature in a puzzle quite as often as it does here. I cast about to thinking what else 'ago' was a substring of, and the word 'dragon' came to mind. Looking at the bottom row of Group 1 (tin / air / ago), something just clicked for me that all three of these appeared in the names of the related Pokémon Dratini, Dragonair and Dragonite (possibly aided by having replayed the game while off sick this week with Covid!), and upon noting the presence of Machop, Machoke and Machamp in the row just above, I realised I was on to something!
It was then just a case of working my way through the various lists...
