In our age of cocooning, it can be hard to remember that real-life Democrats and Republicans are not caricatures like you see on the talking head panel for MSNBC or Fox News. In fact, if both of those channels are skeptical of a proposal, you can bet that there is actually a good deal of support for it among the flesh-and-blood Democrats and Republicans who live around you. The divide in that case is between globalist elites and America-first populists.
The Reuters tracking polls include one about a temporary ban on all Muslims entering the country, which is more extreme than what Trump is proposing -- not on all Muslims, but those from countries with a proven record of producing Islamic terrorists.
As you can see by clicking that link, being against the ban had been comfortably more popular than being for it, from May 17 when they started asking it, until June 5, when it abruptly reverses. For some reason, Reuters stopped asking the question at that point, but put it back in after the Orlando shooting, showing little change from June 6. Presumably by the end of the week, there will be even greater support.
I have no idea what took place around June 5 that reversed the popularity of "for" vs. "against," but it's worth noting that the ban had more for than against already by the time of the Orlando attack.
The other big recent change is an increase of people who say they're merely "not sure".
Putting it all together, first some of those against switched abruptly to being for it (around June 6), and one week later some of those newly "for" it decided to walk it back a bit and said they were just "not sure". The end result is unchanged, though: it's more popular than not, and those who are sitting on the fence are probably for it but just don't want to admit it, or want a slightly watered-down version of it, like Trump has actually proposed.
If you filter the data to look just at Republicans, conservatives, or those who voted for Romney, there hasn't been much change over time -- support is consistently high. The change is entirely due to Democrats, liberals, and those who voted for Obama re-evaluating how wise it is to have unchecked immigration from Muslim countries in an age when they're likely to blow up our leisure spaces because they hate the "decadent West".
Women also show a major change in favor of the ban, while men have remained about equally for and against. White Democrats show a yuge increase in being "for" the ban, while black Democrats remain unchanged. Islamic terrorism is a potential wedge to drive between white and non-white Democrats.
The common denominator of the change in sentiment is those who feel most vulnerable to violence, or who prioritize protection from violence over other matters (such as multiculturalism). That is what liberal morality usually boils down to -- avoiding harm.
But perhaps in this case they're prioritizing culture and lifestyles, and wanting to maintain those against Muslim immigrants who want to impose sharia law on Americans. Whether or not they've read the data from a Pew survey of the world's Muslims, they can tell that most of the Muslims coming here don't want to assimilate to our norms, and want the backward severe laws of their homelands. When over 70, 80, and 90% of a country wants sharia law, "moderate Muslims" are like a needle in a haystack.
Sooner or later, liberals and Democrats will realize that unchecked Muslim immigration will become their worst nightmare, as a puritanical theocracy replaces a secular society.
Disturbingly, though, there is one conservative group that is fiercely against the Muslim ban -- try to take a guess, based on which state Trump lost by the biggest margin, and who is the only lingering high-profile "Never Trump" politician. That's right -- it's the damn Mormons again. The sooner that the Trump-influenced GOP can trade the Mormons for Michigan, the better.
During the entire period of the question, Mormons were 61% against the ban, 27% in favor, and 12% not sure. Other religious groups are either evenly for and against (like the Methodists), or slightly in favor (like the Catholics). It's only the Mormons who are so bitterly opposed to the idea. It stems from their still burning sense of persecution as a religious minority by the Federal Gubmint, which refused to allow Utah to enter the Union until they ended the most backward and whackjob practices of their new-age out-West cult, like polygamy.
That's my guess as to why Romney is still so triggered and tantrum-throwing about the rise of Trump, and crying about the Muslim ban. He's not like the other Establishment Republicans, who are either falling in line or trying to appear neutral. He's a high-ranking official of the Mormon Church, with divided loyalties between it and the nation:
Romney is no run-of-the-mill churchgoer but has held responsible posts in this clergy-less denomination that’s led locally by laymen serving part-time. He has been the “bishop” (equivalent of a pastor) in his own “ward” (congregation) and president of the Boston area “stake” (akin to a Catholic or Episcopal bishop). He is an ordained “high priest,” the LDS ecclesiastical rank below patriarch, seventy, and apostle.
Jews are the other "religious" group that is far more against the ban than for it, being globalist managerial types. But even they are not as naive as the Mormons, and have a healthy minority who want to keep Muslims out of Jewish neighborhoods.
The Trump movement's goal should be to put out feelers for common-sense liberals and Democrats (not progressive activists concerned with signaling their values), and encourage their skepticism of unchecked immigration from around the world, particularly from countries whose immigrants want to impose sharia law on us at a minimum, and occasionally blow up our leisure spaces.
It's also important to distance ourselves from the old Mormon and Mormon-ish base of the GOP, which will soon defect to the Democrats or form a breakaway party, in either case siding with globalist multiculturalism rather than putting America first.