How does egg hatching distance relate to the hatched Pokémon?

Solution 1:

All first evolution stage pokemon except for legendaries can appear. This is spread throughout the eggs as follows:

Chart of egg distributions

You cannot get any further evolution stages of pokemon from an egg, nor can you get a legendary from an egg.

These distances were minimum distances at the start, however, that appears to have been fixed

Solution 2:

2 KILOMETERS

  • Bulbasaur
  • Charmander
  • Squirtle
  • Caterpie
  • Weedle
  • Pidgey
  • Rattata
  • Spearow
  • Zubat
  • Geodude
  • Magikarp
  • Cleffa
  • Igglybuff

5 KILOMETERS

  • Ekans
  • Sandshrew
  • Nidoran(f)
  • Nidoran(m)
  • Vulpix
  • Oddish
  • Paras
  • Venonat
  • Meowth
  • Psyduck
  • Mankey
  • Growlithe
  • Poliwag
  • Abra
  • Machop
  • Bellsprout
  • Tentacool
  • Ponyta
  • Slowpoke
  • Magnemite
  • Farfetch'd
  • Doduo
  • Seel
  • Grimer
  • Shellder
  • Gastly
  • Drowzee
  • Krabby
  • Voltorb
  • Exeggute
  • Cubone
  • Lickitung
  • Koffing
  • Rhyhorn
  • Tangela
  • Kangaskhan
  • Horsea
  • Goldeen
  • Staryu
  • Mr. Mime
  • Tauros
  • Porygon
  • Togepi
  • Pichu

10 KILOMETERS

  • Onix
  • Hitmonlee
  • Hitmonchan
  • Chansey
  • Scyther
  • Pinsir
  • Lapras
  • Eevee
  • Omanyte
  • Kabuto
  • Aerodactyl
  • Snorlax
  • Dratini
  • Magby
  • Smoochum
  • Elekid

Solution 3:

I've found an updated version of what you get for every 2k, 5k, 7k and 10k egg.

You can also view the updated eggs on the Silph Road here

I plan to keep this answer updated when the eggs change.

Feb 1 2019

  • Taillow added to 2k eggs
  • Lileep, Anorith, Feebas, Beldum, Nincada, Absol, Mawile removed from 7k eggs

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Jan 16 2019

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Update

The eggs change so frequently that it's difficult to keep up with replacing the answer here. So please refer to the Silph Road for the updated list.