Once in the United States, if a claim of asylum is made, the family is given paperwork called a Notice to Appear, which explains they are to go to Immigration Court. Many of them never show up in court and are ordered removed from the United States in absentia. Others file their asylum applications and have a court hearing. If denied, an order of removal is the result. In other words, the family has an order of deportation.
You may think that an order of removal means the immigrant will have to leave the United States, but that is not the way it works. If the immigrant did not appear in court, a Motion to Reopen the case may be filed, or if there was an appearance, some other form of relief may be requested such as Prosecutorial Discretion which is an application to remain in the U.S. even though a deportation order has been entered. The removal order may be appealed, and this process can take 7 months or longer. If the appeal is denied, a Motion to Reopen or to Reconsider may be filed taking additional time.
Once the legal process is completed, the Department of Homeland Security simply does not have the resources to round everyone up and send them back. Instead, as of a few years ago, DHS has a policy of focusing on immigrants who have a criminal record.
To try to stem the tide from Mexico, the Bush administration used highly publicized raids on high profile businesses, and now the Obama administration has a new detention policy, which is to actually deport those from Central America who have deportation orders. It remains to be seen how extensive this effort will be or how long it will last. More than 12,000 people were apprehended at the border in October and November, compared with approximately 4,500 in the same months a year ago. since the administration is also trying to increase the number of immigrants who can remain in the United States through Executive Orders, this may be a short lived program once the numbers are reduced.
Of course, the overriding questions are about what reform laws, if any, should be passed, and what will happen to the millions of people in the U.S. without legal status. The emphasis on family reunification should be changed to a combination of family reunification and a point system which encourages immigration of those with education, capital and skills which will help the U.S. economy, and at the same time retain diversity of nationalities. The way to do this is not through comprehensive reform with the current political environment. Instead, you would hope and think that member of Congress could find a way to agree on some steps to take to begin the process of resolving the immigration issues we have.