Yes, some illegal immigrants are bused from the border to various US cities, being released in the US on their own recognizance.
The Houston Chronicle reports (in a subscribers-only article):
A little before noon on Friday, a white Homeland Security bus stopped outside a public bus terminal in downtown McAllen, its doors opening to disgorge a group of about 20 immigrants from Central and South America.
Similarly the San Antonio Express-News reports much the same thing (again, subscribers only):
For weeks now, the Border Patrol has been dropping off hundreds of Central Americans at bus stations in the Rio Grande Valley as an unprecedented wave of immigrants pours across the southern tip of Texas.
The caption to a photograph on that article states that the reason for these immigrants being released is that:
A wave of Central American adults with children and unaccompanied minors has overwhelmed U.S. Immigration and Customs detention centers. Immigration officials release some of them on their own recognizance after undergoing processing.
Some were transported at least as far as California. XETV-TDT reports:
The dropping off of undocumented immigrants from Texas began about eight months ago. At that point, the Border Patrol was either flying them or bussing them to Greyhound stations in Tucson and Phoenix. Even though they're both sizable cities, employees at the bus stations said they were being overwhelmed. Then, for reasons that aren't yet clear, the Border Patrol decided to start taking the immigrants to the Border Patrol station in Murrieta for processing.
"It's called catch and release. We are out there catching people... they are arrested, sorry they are surrendering to agents in the Rio Grande Valley (in Texas) because they know they will be detained for maybe a few days, a few weeks at most and then they will be released," said Agent Shawn Moran, who is with the National Border Patrol Council, the union for Border Patrol agents.
The immigrants are released and then given a date to show up in court. Agent Moran said that rarely happens. And he told San Diego 6 that Border Patrol stations in Murrieta and across the southwest are being overwhelmed. "Many of these stations have ground to a halt in terms of operations because they are doing nothing but processing," said Moran.
The Fox News article you provided claims that "other than Mexicans" are being treated in this manner, which appears to be true. Illegal Mexicans can be and are simply and easily passed back over the border, as the VICE article you provided claims. On the other hand, illegal immigrants from other countries are not so easy to deal with. As the Washington Post reports:
Unlike illegal migrants from Mexico, who can be quickly processed and returned by bus to Mexican border cities, Central Americans cannot be easily shipped home. Airplanes must be chartered. Consular arrangements must be made. And if migrants request asylum in the United States, the U.S. government has the additional responsibility to determine whether their appeal is based on a legitimate need for protection and a “credible fear” of persecution in their home countries.
More than 36,000 migrants, the majority from Guatemala, El Salvador and Honduras, requested asylum along the southwestern border during the government’s 2013 fiscal year, nearly triple the 2012 number. Ultimately, most of the applications were denied, but critics of the process say migrants are gaming the system to extend their stays in the United States. Others may simply go underground and ignore deportation orders if their petitions are rejected.