I didn't know they were refusing to hear asylum report. From what I read they were and those granted asylum were reunited with the children. I also heard they were reunited as soon as the asylum was rejected as well.
What would you do different Morally? You have to make a decision after all. Remember I am not saying this is ideal. This sounds like a mess but I can't think of a better solution myself.
I mean if I was a right-wing ruler with a hatred for outsiders, I could order the governments' agents to do the same without breaking up families. Put them all in the same cages (which are officially not prisons) as the kids. If I was a centrist, I could probably make some miserable temporary housing.
But I'm not, so...
1. Change in domestic drug policy: legalisation of weed and reducing sentences/types of offence for others. Use Portugal as a model, failing that, something like Denmark.
2. Obviously this is not a one-size-fits-all solution, but allowing/giving technical or economic support for other countries within the US' direct sphere of influence (Central America) to do something similar. The US has historically done the opposite.
Both these actions should reduce the violence affected with drugs.
3. Bolder and less morally easy actions wrt cartels - the Mexican presidential frontrunner has proposed amnesty, after decades of "killing the kingpins" strategy backfired totally. Think of them like Kissinger - a mass murderer who will never face justice/mass murders whose deserved punishment will result in more death.
4. I know that Mexican agriculture has been hollowed out in terms of employment since NAFTA, which has partly provided fodder for the drug trade/war. I wouldn't be surprised if this was the case elsewhere too. I think this, alongside the war on drugs, is one of the root causes of both the violence and immigration. Therefore, something along the lines of a (much smaller) Marshall plan to create employment in Central American countries (which should also stimulate some US manufacturing).
5. About the people who entered the US without a visa: unless they've committed violent crimes, give them a path to US citizenship. About the people at the border now - evaluate their claims quickly and leniently. While that is happening - resettlement a. with family members if they have any b. in willing homes c. temporary shelters, not cages staffed by police.
Finally, this is something I've put together from memory, 5 mins of googling, and an embarrassingly low knowledge of specifics. I'm sure there are better-written and better-researched proposals.