In the US, we have a national crisis of a lack of empathy
Taking your last point first, I would say that the US is one of the most empathetic nations on the planet.
The US takes in more immigrants per year than any other country, it allows protests for everything from gay rights to black rights to abortion rights, it allows every religion under the sun and so on.
Of course we could argue that the US gets involved in too many wars that are none of its business and that kill lots of innocent citizens, but then so do many other countries for geopolitical reasons.
I realise though that you are talking more about the empathy of individuals, which I will address below, sort of.
I was thinking about this in the shower, so I'm not sure if this is a coherent idea or just a "shower thought" but I wonder if the BLM protests won't turn out effective because it's aimed at adults. And, by the time you are adult, if you don't already see things from the point of view of different races, sexes, nationalities, etc., it's really hard to start doing it.
I think protests *do* have the ability to energize and mobilize, but I'm not sure they can change hearts and minds because people are able to go to incredibly lengths to rationalize mistreatment and keep the status quo if it suits them.
So, I started thinking (it was a long shower) about how realistic ways to improve justice and equality for all. I think it starts with the arts: fiction, movies, music, visual arts, etc and it starts at a young age.
I remember a literature professor in college telling us that one of the reasons literature is so important is because it's one way we learn empathy. Having to view the world through different characters with different genders, nationalities, races and backgrounds makes us more empathetic. We see the world as they do while we read their story and we care about it.
In my generation, Gen X, we didn't have many heroes that were not white men or boys but it was just starting to change. We had Ripley from the Alien franchise and she was a total badass (one of my favorite Hollywood stories is that the only reason Sigourney Weaver got the part is that she didn't realize it was supposed to be a male lead when she auditioned) and the Cosby Show, which showed a successful black family.
Anyway, I believe this is the reason late Gen X and younger are much less prejudice on the whole and it boils down to learned empathy from having more diverse representation.
I think this is related to tinnitus as well. People with no or mild tinnitus can't imagine how disabling it can be because they have trouble putting themselves in the frame of mind of people dealing with things they can't relate to. It's one reason why people with invisible disabilities are dismissed and told to "toughen" up.
In the US, we have a national crisis of a lack of empathy but hopefully the younger generations will save us -- if they show up to vote.
First, I would question whether empathy for others is always a good thing. A lack of empathy for others/outsiders is sometimes the glue that helps a society keep together and thrive, ie help those like ourselves and not outsiders (whether it be the colour of one's skin, religious beliefs, political beliefs and so on). Take Japan for example where the society is linguistically, ethnically and culturally homogeneous, ie around 98% of the people are ethnically Japanese (although there are three minorities included in that). It has low immigration and places little importance on trying to assimilate foreigners into its society, the same goes for minorities. One could argue that due to this, or partly due to this, Japan is doing rather well, eg economic powerhouse, very long life expectancy, very low levels of violent crime, religion is dominated by Shintoism and Buddhism, democratic political system* and so on. Japan is very nationalistic, and people follow a pretty strict social code. Harmony and social obedience are highly valued. Interestingly though, there is quite a bit of corruption.
In contrast, it could be argued, the US is more and more being torn apart by multiple narratives from opposing forces that have little empathy for each other. Even though the US has done very very well in many areas, which it could be argued is due to its diversity, it is dealing with massive social problems, relatively poor life expectancy, massive crime rates and a total lack of harmony.
* Not sure about how much having a democratic political system makes a difference, but I included it anyway. South Korea could be compared to Japan and so could North Korea, but the latter is doing worse off in many ways using the criteria above.
Second, I feel that most people have an inbuilt 'stranger danger' mechanism that automatically sways them to people similar to themselves, eg conservatives will naturally search out other conservatives, liberals the same, gay people the same, heterosexuals the same, people of different colours the same, people with the same interests and so on. In contrast, some people are born more empathetic to outsiders than others. For example, take me (not that I consider myself the poster boy for empathy), living half my adult life abroad in a country that for a long time has been the enemy of my own, marrying a foreign citizen, learning the country's language, understanding the mentality and so on. I have empathy for these people and for this country in some ways, ie I can see where these people are coming from and how they look at life (this is a very big generalisation of course as there are many points of view among people). But it doesn't mean that I agree with much of what goes on here. But, yes, I have empathy. One that is naturally built in, one that swayed me to go out and see how others live and study anthropology and people and society and culture.
How many people who are white have many or any black friends and vice versa? How many heterosexuals have many or any gay friends and vice versa? How many conservatives have close liberal friends and vice versa? How many close foreign friends do you (not you FGG in particular) have? How many poor people have rich friends and vice versa? As an educated guess, I would say not many to all of the above in relation to most people. Am I wrong?
People, in general, find comfort in similarity, so emphathy for outsiders is not something of a high priority for many. Am I wrong?
Third, although at the beginning I said that I consider the US to be one of the most empathetic countries on the planet, that does not mean everybody within the country is very empathetic. I feel that the nation state US, as a world power, is trying to lead the way and trying to force its own vision of what society should be onto others.
Fourth, can people be made more empathetic? Yes, at least some people, but probably not all. Like you stated films and books and other media etc can definitely sway people's outlooks. You could call it propaganda I suppose, and the use of propaganda is well known to work. And like you said, if you want to make people more empathetic then the earlier in life they start to view other people from their (the other people's perspective) the better.
Will the BLM movement change anything? I'm not so sure. It may cause more divison than actually bring people together. I don't know enough about the movement to judge how well it is doing at achieving its perceived goals. Whatever, something has to happen in the US as at present things are getting to the point where a civil war could become a not so far-fetched possibility.