It does vary with distance.
0 Ring Portals (For colors other than white, these portals always give 3% chance of a legendary if one is possible, and 1% chance of shiny if it's not legendary, regardless of distance.)
605-999: 60% Nihilego, 35% Buzzwole/Pheromosa, 2% Xurkitree, 2% Kartana/Celesteela, 1% Guzzlord
1000-1499: 54% Nihilego, 39% Buzzwole/Pheromosa, 4% Xurkitree, 2% Kartana/Celesteela, 1% Guzzlord
1500-1999: 48% Nihilego, 43% Buzzwole/Pheromosa, 6% Xurkitree, 2% Kartana/Celesteela, 1% Guzzlord
2000-2499: 42% Nihilego, 47% Buzzwole/Pheromosa, 8% Xurkitree, 2% Kartana/Celesteela, 1% Guzzlord
2500-2999: 36% Nihilego, 51% Buzzwole/Pheromosa, 10% Xurkitree, 2% Kartana/Celesteela, 1% Guzzlord
3000-3499: 30% Nihilego, 55% Buzzwole/Pheromosa, 12% Xurkitree, 2% Kartana/Celesteela, 1% Guzzlord
3500-3999: 24% Nihilego, 59% Buzzwole/Pheromosa, 14% Xurkitree, 2% Kartana/Celesteela, 1% Guzzlord
4000-4499: 18% Nihilego, 63% Buzzwole/Pheromosa, 16% Xurkitree, 2% Kartana/Celesteela, 1% Guzzlord
4500-4999: 12% Nihilego, 67% Buzzwole/Pheromosa, 18% Xurkitree, 2% Kartana/Celesteela, 1% Guzzlord
5000+: 6% Nihilego, 71% Buzzwole/Pheromosa, 20% Xurkitree, 2% Kartana/Celesteela, 1% Guzzlord
Buzzwole and Kartana are Sun-exclusive, while Pheromosa and Celesteela are Moon-exclusive. If the odds for one of those slots come up lucky, the world you'll go to depends on the version you're playing.
1 Ring Portals
1000-1499: (white) 2% Nihilego, 54% Buzzwole/Pheromosa, 39% Xurkitree, 4% Kartana/Celesteela, 1% Guzzlord; (other colors) 4% legendary, 2% shiny if it's not legendary
1500-1999: (white) 2% Nihilego, 48% Buzzwole/Pheromosa, 43% Xurkitree, 6% Kartana/Celesteela, 1% Guzzlord; (other colors) 5% legendary, 3% shiny if it's not legendary
2000-2499: (white) 2% Nihilego, 42% Buzzwole/Pheromosa, 47% Xurkitree, 8% Kartana/Celesteela, 1% Guzzlord; (other colors) 6% legendary, 4% shiny if it's not legendary
2500-2999: (white) 2% Nihilego, 36% Buzzwole/Pheromosa, 51% Xurkitree, 10% Kartana/Celesteela, 1% Guzzlord; (other colors) 7% legendary, 5% shiny if it's not legendary
3000-3499: (white) 2% Nihilego, 30% Buzzwole/Pheromosa, 55% Xurkitree, 12% Kartana/Celesteela, 1% Guzzlord; (other colors) 8% legendary, 6% shiny if it's not legendary
3500-3999: (white) 2% Nihilego, 24% Buzzwole/Pheromosa, 59% Xurkitree, 14% Kartana/Celesteela, 1% Guzzlord; (other colors) 9% legendary, 7% shiny if it's not legendary
4000-4499: (white) 2% Nihilego, 18% Buzzwole/Pheromosa, 63% Xurkitree, 16% Kartana/Celesteela, 1% Guzzlord; (other colors) 10% legendary, 8% shiny if it's not legendary
4500-4999: (white) 2% Nihilego, 12% Buzzwole/Pheromosa, 67% Xurkitree, 18% Kartana/Celesteela, 1% Guzzlord; (other colors) 11% legendary, 9% shiny if it's not legendary
5000+: (white) 2% Nihilego, 6% Buzzwole/Pheromosa, 71% Xurkitree, 20% Kartana/Celesteela, 1% Guzzlord; (other colors) 12% legendary, 10% shiny if it's not legendary
2 Ring Portals
1000-1499: (white) 0% Nihilego, 2% Buzzwole/Pheromosa, 54% Xurkitree, 39% Kartana/Celesteela, 5% Guzzlord; (other colors) 10% legendary, 3% shiny if it's not legendary
1500-1999: (white) 0% Nihilego, 2% Buzzwole/Pheromosa, 48% Xurkitree, 43% Kartana/Celesteela, 7% Guzzlord; (other colors) 13% legendary, 5% shiny if it's not legendary
2000-2499: (white) 0% Nihilego, 2% Buzzwole/Pheromosa, 42% Xurkitree, 47% Kartana/Celesteela, 9% Guzzlord; (other colors) 16% legendary, 7% shiny if it's not legendary
2500-2999: (white) 0% Nihilego, 2% Buzzwole/Pheromosa, 36% Xurkitree, 51% Kartana/Celesteela, 11% Guzzlord; (other colors) 19% legendary, 9% shiny if it's not legendary
3000-3499: (white) 0% Nihilego, 2% Buzzwole/Pheromosa, 30% Xurkitree, 55% Kartana/Celesteela, 13% Guzzlord; (other colors) 22% legendary, 11% shiny if it's not legendary
3500-3999: (white) 0% Nihilego, 2% Buzzwole/Pheromosa, 24% Xurkitree, 59% Kartana/Celesteela, 15% Guzzlord; (other colors) 25% legendary, 13% shiny if it's not legendary
4000-4499: (white) 0% Nihilego, 2% Buzzwole/Pheromosa, 18% Xurkitree, 63% Kartana/Celesteela, 17% Guzzlord; (other colors) 28% legendary, 15% shiny if it's not legendary
4500-4999: (white) 0% Nihilego, 2% Buzzwole/Pheromosa, 12% Xurkitree, 67% Kartana/Celesteela, 19% Guzzlord; (other colors) 31% legendary, 17% shiny if it's not legendary
5000+: (white) 0% Nihilego, 2% Buzzwole/Pheromosa, 6% Xurkitree, 71% Kartana/Celesteela, 21% Guzzlord; (other colors) 34% legendary, 19% shiny if it's not legendary
2 Ring + Bloom Portals (White portals in this style can never go to Nihilego, Buzzwole, Pheromosa, or Xurkitree's world regardless of distance. Other colored portals have a 100% chance of producing a legendary encounter if there are any eligible legends left in that color, so the "shiny if it's not legendary" odds can only apply if there are in fact no such legends eligible--either you've caught them all, or the only ones you haven't caught yet are "third members" of legendary trios where they're configured to only be eligible to appear if you bring the other two members of that trio in your party along for the warp ride.)
2500-2999: (white) 40% Kartana/Celesteela, 60% Guzzlord; (other colors) 16% shiny if it's not legendary
3000-3499: (white) 34% Kartana/Celesteela, 66% Guzzlord; (other colors) 20% shiny if it's not legendary
3500-3999: (white) 28% Kartana/Celesteela, 72% Guzzlord; (other colors) 24% shiny if it's not legendary
4000-4499: (white) 22% Kartana/Celesteela, 78% Guzzlord; (other colors) 28% shiny if it's not legendary
4500-4999: (white) 16% Kartana/Celesteela, 84% Guzzlord; (other colors) 32% shiny if it's not legendary
5000+: (white) 10% Kartana/Celesteela, 90% Guzzlord; (other colors) 36% shiny if it's not legendary
