For example, 350.org is a 501(c)(3) non-profit organization (according to its footer, bottom right) but signing up for its e-mail list or otherwise supporting it means a person would also wind up getting e-mails from "350 Action" (with list management still hosted at 350.org) making direct statements including "Vote for Hillary Clinton" and trying to recruit folks into canvassing on behalf of a particular candidate.
A key part of 350.org's methodology is "Pressure governments into limiting emissions" including "Run local and national campaigns targeting leaders." 350.org staff include people with job titles like "Campaigner," "Campaign Coordinator/Manager/Director," "US National Field Political Organizer," etc.
According to the IRS in 2016,
For an organization to be tax-exempt under section 501(c)(3) it cannot “participate in, or intervene in (including the publishing or distributing of statements) any political campaign on behalf of (or in opposition to) any candidate for public office.”
Elsewhere, the IRS states more directly:
"Under the Internal Revenue Code, all section 501(c)(3) organizations are absolutely prohibited from directly or indirectly participating in, or intervening in, any political campaign on behalf of (or in opposition to) any candidate for elective public office. The prohibition applies to all campaigns including campaigns at the federal, state and local level...Political campaign intervention includes any and all activities that favor or oppose one or more candidates for public office. The prohibition extends beyond candidate endorsements. Contributions to political campaign funds or public statements of position (verbal or written) made by or on behalf of an organization in favor of or in opposition to any candidate for public office clearly violate the prohibition on political campaign intervention. Distributing statements prepared by others that favor or oppose any candidate for public office will also violate the prohibition. Allowing a candidate to use an organization’s assets or facilities will also violate the prohibition if other candidates are not given an equivalent opportunity."
There are also conservative charitable organizations supporting Trump, churches that may have slightly different rules, charities bearing the names of each candidate, etc.; this question is not limited to 350 but 350 just served as a conveniently concrete and sufficiently transparent example to help illustrate the kind of practices being asked about in this question.
What is the minimum degree of separation between a 501(c)(3) and its affiliated 501(c)(4) organization? Is it sufficient that a difference exist on paper (e.g. different articles of incorporation, different domain name, different bank account)? Are there restrictions against staff/directorial overlaps or the 501(c)(3) contributing its database/mailing list, etc. to its 501(c)(4) (including those that may have been satisfied behind the scenes)?
SE site selection notes: I recognize this might also be on topic at Law.SE, but I believe it's also on-topic here and this site has more specific domain experts who might be more knowledgeable about these specific laws. There seems to be a rule that a question asked on a site with a more general domain (like Law) about a specific topic that another SE exists for (like Politics), it would be closed on the former site without migration to the latter.