Word to describe a group of people helping each other based on kinship

The generic term for such a group is literally "kinship group" which Oxford defines as

kinship group [noun] Anthropology: A family, clan, or other group based on kinship (...) Example: a clan is a corporate kinship group whose members are considered blood relatives and claim descent from a common ancestral spirit, but who may not know their precise genealogical relationship.

Source: https://en.oxforddictionaries.com/definition/kinship_group (see "example sentences")

Specifically, the term clan seems to fit your meaning:

1A close-knit group of interrelated families, especially in the Scottish Highlands

1.1 A large family

1.2 A group of people with a strong common interest.

Source: https://en.oxforddictionaries.com/definition/clan

In addition,

a descent group is a social group whose members talk about common ancestry.

Source: https://en.m.wikipedia.org/wiki/Kinship

In India we simply say "relatives" or "fellow-villagers" which means anyone related by kinship as in blood or marriage, or belonging to the same village/town/region. Even a remote relative or an unknown fellow-villager can have a claim on help and support in time of need, in communities that subscribe to that concept.

Example sentences that fit your context:

When the Indian student faced a financial crunch in New York, he was substantially supported by members of his kinship group, including relatives and fellow-villagers living in the USA.

In his time of need, Mike Smith got help from his uncle's brother-in-law working in the same city: the Smith clan looks out for each other.

NOTE: The word "clan" originated in the Scottish Highlands but is now used universally to denote a kinship group, as demonstrated by the many example sentences at Oxford Dictionaries Dot Com.

I believe that the next tier up from one's own family (in respect of courteous regard and charitable assistance) is the local community.

A body of people who live in the same place, usually sharing a common cultural or ethnic identity. Hence: a place where a particular body of people lives.

OED

This is just one of the OED's definitions as it defines 'community' under several types, whether geographical, ethnic, religious, cultural or ideological. But whatever the precise realm in which the community operates (in terms of the OP - whatever type of kinship is operating) I think that 'community' expresses the concept best.

a sept TFD

A division of a family, especially a division of a clan.

As in (sentence provide by me):

This close knit sept kept watch on each other's needs.

Names for organized groups of such a type:

An organized group of people created to assist people based on kinship could be called an "Ethnic Benefit Society" - frequently these might have been mens clubs in the past... "Service Societies" etc

See : Benefit Society at Wikipedia

A benefit society, fraternal benefit society or fraternal benefit order is a society, an organization or a voluntary association formed to provide mutual aid, benefit, for instance insurance for relief from sundry difficulties. Such organizations may be formally organized with charters and established customs, or may arise ad hoc to meet unique needs of a particular time and place.

....

Benefit societies may be organized around a shared ethnic background, religion, occupation, geographical region or other basis. Benefits may include financial security and/or assistance for education, unemployment, birth of a baby, sickness and medical expenses, retirement and funerals. Often benefit societies provide a social or educational framework for members and their families to support each other and contribute to the wider community.

Words for the behavior of a group helping members of their own group ?

If it is a term to describe the behavior of ethnic mutual aid, try the terms discussed in the Wikipedia articles below.

Take care in discussing these group dynamics as any matter of race, religion or generalizing a behavior is always a touchy subject. These are sociology terms, that might belong more appropriately in a context that has invited such examination.

Ethnic Nepotism at the Wikipedia

In sociology, the term ethnic nepotism describes a human tendency for in-group bias or in-group favoritism applied by nepotism for people with the same ethnicity within a multi-ethnic society.

The term was coined in the 1960s in the context of the ethnic (tribal) tensions and rivalry in the then-recently independent states in Sub-Saharan Africa such as Nigeria.2

... it is a long article with many terms and theories referenced with further links

related:

In-group Favortism at the Wikipedia

in-group favoritism, sometimes known as in-group–out-group bias, in-group bias, or intergroup bias, is a pattern of favoring members of one's in-group over out-group members. This can be expressed in evaluation of others, in allocation of resources, and in many other ways.

This interaction has been researched by many psychologists and linked to many theories related to group conflict and prejudice. The phenomenon is primarily viewed from a social psychology standpoint. Studies have shown that in-group favoritism arises as a result of the formation of cultural groups.3[4] These cultural groups can be divided based off seemingly trivial observable traits, but with time populations grow to associate

... another long article that should be consulted before throwing the term around