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Rep. Tom Cole, Oklahoma Republican, said military bases were always bad fits for both sides. Legislation against the use of military bases for sheltering illegal immigrants passed the House last week as part of a package of bills designed to halt the surge of children and families.

Senators have failed to pass their own bill, however, leaving Mr. Obama to manage the situation through August without any new resources or legal powers.

Republicans said the key to stopping the surge is to change a 2008 law and make it easier to deport the children from Central America.

Under the law and the Obama administration’s interpretation, the children are given extensive screenings, sent to the shelters and then connected with relatives living in the U.S. — often illegally — where they are to await immigration court proceedings.

Some of the children fail to show for those proceedings and disappear into the population, while others win legal status. Both scenarios act as incentives for more children to cross, Republicans say.

Democrats argue that the surge of children is small enough for the U.S. to handle and that the situation should be considered a humanitarian crisis rather than a border problem.