It shouldn’t. If a congregation doesn’t like it, it can drop it’s affiliation with the movement, but if it wants to maintain it, it complies with its policies.
The movements don’t go out to areas where there isn’t a congregation, and start a mission that is supported financially by the national movement until it can support itself, with the goal of recruiting members in order to become self-supporting.
Congregations start because there are a lot of Jews in an area who start coming together and decide they are a congregation; they draw up a charter or a constitution. They may build or buy a building. They may decide to affiliate with one movement, more than one, or none.
My current congregation is affiliated with both the Conservative and the Reconstructionist movements. The last synagogue I belonged to was affiliated with no movement for more than 20 years, but at one point, after being without a rabbi for a few years, decided to affiliate with the Reform movement in order to host student rabbis, and have them there every Shabbat, and on the holidays to perform some rabbinical functions.
However, there are “musts,” and “permitteds.” In other words, things that a congregation is required to do, or to recognize, and things that the congregation is allowed to do or recognize.
Recognizing same-sex marriages is required of all three liberal movements. This means that anything that is extended to a heterosexual married couple, in terms of joint memberships, or whatever, MUST also be extended to same-sex couples who are legally married.
I think the current policy in all the liberal movements is that the rabbi MUST perform a marriage ceremony for any couple where both people are Jewish, at least one is a member of the congregation, they have a valid license in the jurisdiction, and there is no legal reason they should not be married (for example, if one is divorced, a get has been issued). However, I have not checked to make sure that this is the policy for the Conservative movement. I would need to check, because there was a time when rabbis were allowed to refuse same-sex couples-- it was when it was not legal in all states.
The movements did not all come along equally. The Reconstructionist movement allowed rabbis to say a blessing over same-sex couple IIRC, back in the 1980s, and Reform and Reconstructionist movements allowed congregations to choose to offer joint memberships to same-sex couples in the 1990s.
I’m pretty sure that since same-sex marriage has been allowed in the whole country, it has been required of all rabbis in all three liberal movements.
There is a fourth sort of denomination or something, of liberal Judaism called “Renewal.” It’s kind of new-agey, and very touchy-feely. Not my cup of tea at all. But they are very much OK with same-sex marriage. They are pretty new, though, and don’t currently ordain rabbis.