Usually it's calling for more interference of state in church. Limits, taxes, restrictions. At least on mainstream, "we don't like you" churches.
Religious organizations don't have untraceable money. I was bookkeeper for one. It's traceable. The books are public, or at least were.
I personally think that they have the same sort of moral status as unions: They're large free associations of citizens for a specific cause. To this extent, they're citizen-controlled. If a Methodist church wants to put money to a (R) or (D) candidate, the congregation will know; it will approve or disapprove, and take appropriate action. Those who disagree will leave, leaving the church more polarized and with less money; others will be attracted, bringing in more money and even more polarization. I think it's a bad thing, but to each his own.
In the end, the wise church that has no mainstream pretensions will eschew meddling too much in politics. After all, look at the unions: It's brought a lot of bad people to power and led to a lot of corruption, it's brought resentment on the part of many who didn't want to pay for union political support, it's led to unions supporting politicians who pass their legislation (typically viewed as buying the politicians, unless it's for the right cause, then it's a grand and glorious thing, money for legal defense of yours and legal oppression of the enemy). Even when I was a TA union member, I refused to pay the non-obligatory part of the dues, and still knew that the "full time" employees my dues go toward spent a lot of their time coordinating with lobbyists and working on political causes. They'd meet to discuss "union representation" but talk about something entirely different--and who's going to audit the private conversations behind closed doors? Personally, when I want to contribute to a party or campaign, I do so, and would prefer my union or my church stay uninvolved.
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