TMs that have been available since move introduction:
Solar Beam
Hyper Beam
Blizzard (TR in Gen VIII)
Earthquake (TR in Gen VIII)
Psychic (TR in Gen VIII)
Fire Blast (TR in Gen VIII)
Rest
Sunny Day* (introduced in Gen II)
Protect (introduced in Gen II)
Rain Dance* (introduced in Gen II)
Frustration* (introduced in Gen II)
Hidden Power* (introduced in Gen II)
Return* (introduced in Gen II)
Shadow Ball (introduced in Gen II, TR in Gen VIII)
Sludge Bomb (introduced in Gen II, TR in Gen VIII)
Sandstorm* (introduced in Gen II)
Attract* (introduced in Gen II)
Thief* (introduced in Gen II)
Dragon Claw* (introduced in Gen III, TR in Gen VIII)
Calm Mind (introduced in Gen III)
Hail* (introduced in Gen III)
Bulk Up (introduced in Gen III)
Taunt (introduced in Gen III, TR in Gen VIII)
Light Screen (introduced in Gen III)
Safeguard* (introduced in Gen III)
Brick Break (introduced in Gen III)
Flamethrower (introduced in Gen III, TR in Gen VIII)
Rock Tomb* (introduced in Gen III)
Facade (introduced in Gen III)
Overheat* (introduced in Gen III, TR in Gen VIII)
Focus Blast* (introduced in Gen IV, TR in Gen VIII)
Energy Ball* (introduced in Gen IV, TR in Gen VIII)
False Swipe*** (introduced in Gen II, made available as a TM in Gen IV)
Fling* (introduced in Gen IV)
Will-O-Wisp** (introduced in Gen III, made available as a TM in Gen IV)
Embargo* (introduced in Gen IV)
Shadow Claw* (introduced in Gen IV)
Payback* (introduced in Gen IV)
Giga Impact* (introduced in Gen IV)
Stone Edge* (introduced in Gen IV, TR in Gen VIII)
Gyro Ball* (introduced in Gen IV, TR in Gen VIII)
X-Scissor (introduced in Gen IV, TR in Gen VIII)
Poison Jab (introduced in Gen IV, TR in Gen VIII)
Grass Knot* (introduced in Gen IV, TR in Gen VIII)
U-Turn (introduced in Gen IV)
Flash Cannon (introduced in Gen IV, TR in Gen VIII)
Trick Room* (introduced in Gen IV)
Psyshock* (introduced in Gen V, TR in Gen VIII)
Venoshock* (introduced in Gen V)
Sludge Wave* (introduced in Gen V, TR in Gen VIII)
Round* (introduced in Gen V)
Scald (introduced in Gen V, TR in Gen VIII)
Sky Drop* (introduced in Gen V)
Acrobatics* (introduced in Gen V)
Volt Switch* (introduced in Gen V)
Bulldoze* (introduced in Gen V)
Wild Charge* (introduced in Gen V, TR in Gen VIII)
Snarl* (introduced in Gen V)
Dazzling Gleam (introduced in Gen VI, TR in Gen VIII)
Brutal Swing* (introduced in Gen VII)
Smart Strike* (introduced in Gen VII)
Fly** (introduced in Gen I as an HM until Gen VI, made available as a TM in Gen VII)
Surf** (introduced in Gen I as an HM until Gen VI, made available as a TM in Gen VII, TR in Gen VIII)
Waterfall** (introduced in Gen II as an HM until Gen VI, made available as a TM in Gen VII, TR in Gen VIII)
Helping Hand** (introduced in Gen III, made available as a TM in LGP/E)
Foul Play** (introduced in Gen V, made available as a TM in LGP/E, TR in Gen VIII)
Play Rough** (introduced in Gen VI, made available as a TM in LGP/E, TR in Gen VIII)
Superpower** (introduced in Gen III, made available as a TM in LGP/E, TR in Gen VIII)
Drill Run** (introduced in Gen V, made available as a TM in LGP/E, TR in Gen VIII)
Megahorn** (introduced in Gen II, made available as a TM in LGP/E, TR in Gen VIII)
TMs that aren't in one generation:
Toxic (absent from Gen VIII)
Dig (absent from S/M and US/UM
Ice Beam (absent from Gen II, TR in Gen VIII)
Thunderbolt (absent from Gen II, TR in Gen VIII)
Reflect (absent from Gen II)
Steel Wing* (introduced in Gen II, absent from Gen V)
Aerial Ace* (introduced in Gen III, absent from Gen VIII)
Torment* (introduced in Gen III, absent from Gen VIII)
Charge Beam* (introduced in Gen IV, absent from Gen VIII)
Rock Polish* (introduced in Gen IV, absent from Gen VIII)
Dark Pulse (introduced in Gen IV, absent from Gen V, TR in Gen VIII)
Smack Down* (introduced in Gen V, absent from Gen VIII)
Flame Charge* (introduced in Gen V, absent from Gen VIII)
Low Sweep* (introduced in Gen V, absent from Gen VIII)
Echoed Voice* (introduced in Gen V, absent from Gen VIII)
Quash* (introduced in Gen V, absent from Gen VIII)
Retaliate* (introduced in Gen V, absent from Gen VII)
Frost Breath* (introduced in Gen V, absent from Gen VIII)
Dragon Tail (introduced in Gen V, absent from Gen VIII)
Work Up* (introduced in Gen V, absent from Gen VI)
Infestation* (introduced in Gen VI, absent from Gen VIII)
Nature Power*** (introduced in Gen III, made available in Gen VI, absent from Gen VIII)
Confide* (introduced in Gen VI, absent in Gen VIII)
Leech Life** (introduced in Gen I, made available in Gen VII, absent in LGP/E, TR in Gen VIII)
TMs introduced in Gen VII that were absent from Gen VIII were not included.
* Absent from LGP/E and/or unusable in Gen VIII. Still listed because only small selection of moves introduced after Gen I were made available in the game.
** Became a TM after the move was introduced
Sources: List of TMs and list of TRs, as well as checking the availability of certain moves with Bulbapedia's lists for Generation VII and Generation VIII.
