Washington (CNN) The Trump administration was only able to reunite fewer than half of the eligible separated migrant families in its care by a court-ordered deadline Tuesday -- as the fates of thousands more remain in limbo.

A federal judge on Tuesday was nevertheless mostly satisfied with the government's efforts to meet his deadline, turning his attention next to the 2,000 to 3,000 families that will need to be reunited later in the month.

As of late Tuesday afternoon, four children of 102 identified for the deadline had already been reunified with their parents, and another 34 were expected to be reunited by the end of the day, according to the administration. That was fewer than the 54 children the federal government had previously said would be reunited by the deadline.

The children who are addressed under the Tuesday deadline are all under age 5 and have been held by the government for weeks or months after being separated from their parents at the border.

Just under half of those eligible will be reunited on time, and they're just a fraction of the thousands of young immigrants still in custody apart from their parents, many of whom are detained and separated from family as a byproduct of the Trump administration's "zero-tolerance" policy.

Based on information provided in court filings, US District Judge Dana Sabraw said in a court status hearing Tuesday he believes officials should be able to reunite 63 of the kids under age 5 with their parents by the deadline or "within the immediate proximity of today."

The remaining kids would take longer, he acknowledged. Some of their parents were ultimately not eligible for reunification, for reasons which include having a criminal history or not being biologically related to the children. Other parents had already been deported or released and have not yet been located.

But, he said, "these are firm deadlines, not aspirational goals."

Beyond the 38 who were to be reunited by Tuesday, another 16 children were to be reunited "soon thereafter," the filing said, as soon as they are confirmed as eligible, and another eight were flagged for follow-up regarding the safety and suitability of the parent. One more child's parent had an unresolved issue with the background check.

The plan for Tuesday's reunions is for the children to be released to Immigration and Customs Enforcement, which runs the adult detention centers where their parents are being held. Once the children are handed over to their custody, ICE will release the families together with ankle monitors for the parents, an Immigration and Customs Enforcement official told reporters on a conference call. This plan bypasses the lengthy "sponsor" process that is required to release an unaccompanied immigrant child from Health and Human Services custody.

Twelve more kids have parents who cannot be located yet, either because they had already been deported or were released into the country already.

There are 27 children that the government has determined are not eligible for reunification yet, either because the adult is ineligible due to criminal history or not actually being the parent, or because the parents present a danger to the child or have a contagious disease. A handful are also still in criminal custody.

The judge instructed the government attorneys and American Civil Liberties Union attorneys representing the immigrants in the case to discuss the remaining cases to make sure they agree. Any disputed situations -- including if the ACLU does not believe a criminal history is accurate, for example -- are to be referred to the judge in a status update filing due Thursday.

Also due Thursday: information on how many thousands more children aged 5 and older remain to be reunified by his next deadline, July 26.

Sabraw set another hearing for Friday.

Government's 'obligation'

Tuesday's hearing was also forward-looking, as Sabraw largely sided with the ACLU on how to process families moving forward.

The judge repeatedly noted that it was the government's obligation to reunite the families it had separated, and thus the full procedures it would use for children in its care who arrived to the US alone shouldn't apply.

"The idea of an application process doesn't fit in this context. The parent has a right to be reunified and it's the government's obligation to make it so," he said.

Parents, Sabraw said, shouldn't have to prove they're good sponsors for their own children.

The government and ACLU had disagreed over several procedural steps for reuniting parents. Siding with the ACLU in large part, Sabraw said the government should only DNA test families when necessary -- either because of time constraints or a lack of a paper trail.

He also said procedures like requiring parents to file a "sponsor care plan" should not be required before parents get their kids back.

"The parents are not applying. They don't have to prove that they're going to be a good sponsor," Sabraw said. "What the government has to look to is whether the parent is unfit or a danger."

He also noted that already-deported parents may be a "big issue," asking attorneys for both sides to address the issue of how to handle those reunions in their filing later this week.

"This is going to be a big issue it appears," Sabraw said, "because if we have 12 out of 101 or 102, when we look at the next 2,800, 2,900 individuals, I'm assuming there's going to be a commensurate number of individuals who were removed."

Government defends its practices

In an earlier call with reporters, officials from the Department of Health and Human Services as well as Immigration and Customs Enforcement defended the slow pace of family reunifications, citing a number of criminal records and other issues that came up when they checked into parents to meet today's court deadline.

"Our process may not be as quick as some would like, but there is no question that it is protecting children," said Chris Meekins, the chief of staff for the Office of the Assistant Secretary for Preparedness and Response at HHS. "Let me be clear, HHS could have transferred every child out of ORR care to a parent who is in DHS custody today if we did not take into account child safety or if the adult is actually a parent."

He said if HHS was not careful, and had simply reunited all the kids without checks, "we would be putting them in the care of a rapist, a kidnapper, a child abuser and someone who is accused of being a murderer in their home nation."

The court documents and the call noted that in one child's case, for example, another adult in the home had a record of sexually abusing a 10-year-old girl. But the parent will be allowed to choose a different living situation and then apply again to be reunited with their kid.

As for parents who can't be located, the government indicated that the blame falls on the parent.

"There's a very large number of illegal aliens here in the country. A parent that's been released ... always has the capacity to go to HHS and request to sponsor their child out of HHS custody," said Matthew Albence, executive associate director of ICE Enforcement and Removal Operations. "That happens every day. So if these individuals choose not to do that, there's nothing we can do to force them, and it makes me question whether they were the parent in the first place."