People don't leave out the connections, it's just that there isn't much firm evidence to outline whatever nature it had. More difficult statements about relationships outside the circumpolar world remain purely speculative.
As for the follow up questions, Asia is a big place, and India is on the order of 4,000-5,000 miles from the Bering strait where pre-Thule cultures likely originated. There's no identifiable, historic relationship between modern Inuit/eskimos and people in India. However, they remain closely related to the Yuit on the other side of the strait.
Sorry, I meant Indian culture as in American Indian. Native American is confusing in this context as it could refer to Inuits as well.
The Asian culture I’m thinking about would be from Siberia or Kamchatka.
Interestingly, the Ainu people historicly lived pretty close to the Kamchatka peninsula (perhaps even on it), guessing that some culture that later was adopted by the Japanese from the Ainu but was also brought to North-America by proxy of pre-inuit peoples is not that far off.
EDIT: And as for what they brought over, couldn’t the American Indians further south have benefited from escaped domesticated animals like the dog breeds they brought over from Asia. And wouldn’t the closest neighboring tribes have learned different hunting techniques from each other, techniques that would have traveled south. Did pre-inuit people never trade or fight with American Indian people? I’m sure there is a lot of interesting interactions going on there that we are missing.
As for the follow up questions, Asia is a big place, and India is on the order of 4,000-5,000 miles from the Bering strait where pre-Thule cultures likely originated. There's no identifiable, historic relationship between modern Inuit/eskimos and people in India. However, they remain closely related to the Yuit on the other side of the strait.