Trainers » Dust Island | Attacks » Poison Cultivation
If my already-poisoned Pokemon is hit by Dragalge's "Poison Cultivation" attack then I use Switch while Dust Island is in play, is my new Active Pokemon poisoned with a regular condition (only one damage counter in between turns) or with the effect from "Poison Cultivation" (ten damage counters in between turns)?
If the Poison effect places more than one damage counter during the Pokémon Checkup phase between turns, then switching the Pokémon out using a Trainer card while Dust Island is in play will retain the modified Poison effect of placing multiple damage counters on the new Active Pokémon.
Source: TPCi Rules Team (2020-09-24)
Trainers » Dust Island
If I use a Trainer card to switch my Active Pokemon that is affected by Poison from Muk & Alolan Muk GX's "Severe Poison" attack and Dust Island is in play, does my new Active Pokemon take the extra damage counters?
Yes, it does. Dust Island specifies that it is affected by "that Special Condition".
Source: Unbroken Bonds FAQ (2019-04-19), TPCi Rules Team (2019-05-02)
Trainers » Dust Island
If my opponent uses Pokemon Catcher or Escape Rope while Dust Island is in play, is my new Active Pokemon still poisoned.
No, it is not. Dust Island only works when a Trainer card is used to switch your own Pokemon; it does not apply when the Trainer card is played by your opponent.
Source: Unbroken Bonds FAQ (2019-04-19), TPCi Rules Team (2019-05-02)