Liberals as a party focus mainly on the "promote the general Welfare and secure the blessings of Liberty". Conservatives focus more on the "ensure domestic Tranquility [and] provide for the common Defense". But let's look at each of these five goals relative to morality.
If your goal is promoting welfare, then morality isn't much of an issue. People generally agree that feeding people is good, that murder/mayhem in the streets is bad, that paved roads and clean drinking water are good, that driving 90 in a school zone or residential neighborhood is bad, and so on.
For defense, again generally people agree that being safe is good, police (as a concept) are good, a strong military (for defense) is good, buildings blowing up is bad, etc. How much can be sacrificed to achieve these is a different issue (see liberty, below). So again, morality isn't a big factor in these issues.
For "establish Justice", it doesn't matter what your morality is, justice is good. What constitutes justice may vary, but that goes more to the other issues (liberty and tranquility, see below). So again, the concept of wanting justice by itself is not morality-dependent.
Which leaves us with the balance between ensuring domestic tranquility and ensuring the blessings of liberty. Here's where we get interesting.
Conservatives want tranquility, harmony, consistency, even at the expense of some liberty interests. Liberals want greater personal liberties, even if that means letting some criminals go free or not regulating certain behaviors, thus impacting tranquility.
Morality is the set of beliefs that determine what is more important to us. Valuing liberty over tranquility means allowing other people to do things you don't approve of, while holding tranquility over liberty means that you want your values to apply to everyone, in order to avoid conflict.
Is the right to choose who you want to marry (regardless of race or gender) more important than preserving a traditional historical opposite-sex same-race marriage? Until 1967, mixed-race marriages were illegal in many states (mostly conservative ones) because marriage was "by definition" a union between people of the same race. It was a moral issue, because their religions didn't favor mixed-race marriage. The US Supreme Court said that legal restrictions shouldn't be based on religions dogma, and that the right to marry who you want isn't subject to state controls. So far, that ruling only applies relative to race-based decisions, not gender-based decisions.
Is it right for a woman to choose what happens inside her own body? The conservative view is that society as a whole mandates controls, and that based on religious beliefs since the embryo can become a person it should be treated as already being a person. Hence abortion is immoral. However, let's say you don't share the same religious perspective. Let's say you believe that it is the soul that makes someone human, and the soul doesn't show up until the fetus is capable of independent life (viability). In that case, based on your religious beliefs, terminating a pregnancy before the soul enters the body is not murder. How can you prove when the soul enters the body (conception, viability, birth)? You can't. It's a religious issue. So the US Supreme Court went with a medical (not religious) definition of life, and said that states cannot dictate religious definitions as a matter of law.
America is a republic. That means we collectively get together and elect people to make decisions for us. In a pure democracy, might (majority) makes right. In a republic, it's might-makes-right by proxy.
Where there is general consensus, no problem. Killing a person is generally bad. OK. Does 'person' include convicted murderer/rapist? Terrorist? Unborn embryo? Mental retard? Now, the lines get fuzzier.
Convicting and punishing criminals is good; punishing innocent people is bad. OK, so far so good. But would you rather one innocent person go to jail, or one guilty person go free? Or ten? Or a thousand? Or do the numbers change for innocent being convicted versus guilty going free? We're back to that liberty versus tranquility balance.
Defense is good. Preventing attacks is good. Terrorists are bad. Ok, no big disagreement there. But is better to convict people because you think they might someday commit terrorist acts (as just happened in California recently), or wait to see if they actually attempt it? Do we want to punish people by association, because that might lead to a future planned attack? Or allow freedom of association, and simply try to stop the real criminals? Do we hold people indefinitely on suspicion of being terrorists, just in case? Or do we let them go is we can't meet the standards of probable cause necessary to bring criminal charges? Again, which is more important, tranquility or liberty?
That is the balance that society will always need to evaluate. Which is more important, liberty or tranquility? Preventing potential harm by sacrificing freedoms? Or preserving freedoms, even if that means exposing us to risk of harm?
Personally, I think a society can survive risk of harm, and survive lack of tranquility, much better than it can survive loss of liberty. I'd rather live in a society that is free, and works together to keep each other safe, than a tyranny where the government has decided that security means sacrificing freedoms and giving up privacy. "Those who would sacrifice essential liberties in the name of temporary security deserve neither liberty nor security". We as a society can survive terrorists. Can we survive loss of liberty?