What happens to all those asylum-seekers at the Canadian border?

What happens to all those poor people we see trekking across the Prairies or the back end of Quebec in a snowstorm, seeking asylum from the iniquities of today's America in immigrant-friendly Canada? In the first seven weeks of 2017, 4,000 asylum-seekers have applied to Canada from the USA (not all of them presumably on foot overland), compared to 2,500 during the same time frame last year, and the numbers are expected to rise further as the winter starts to subside and Donald Trump starts to impose more restrictions on immigrants and refugees.
Day after day, I see pictures of travel-weary families braving the elements of the Canadian winter, some with a full set of suitcases, some with little more than the clothes they stand up in. Often, I see them being hand-cuffed and led away from border crossings in Emerson, Manitoba or Hemingford, Quebec. But what actually happens to them after the cameraman leaves?
CBC has answered just that question for me. It seems that, most times, the RCMP are aware of the incursions in advance, from their video surveillance footage, and are on hand to receive the would-be refugees. The RCMP is legally obliged to let the travellers know that, if they cross the border, they will be committing an offence under the Canadian Customs Act, and that they risk arrest by proceeding. Handcuffs are not always used; that is a case-by-case decision based on the perceived risks of the individuals involved.
But, either way, arrest is usually exactly what the asylum-seekers want, so that then they can make a refugee claim under the Canada-US Safe Third Country Agreement, something they would not be able to do if they had tried to enter Canada at a legal border entry point (where they would just be turned away). The refugee claimants are then taken into police custody, questioned about their identities, and then handed over to the Canadian Border Services Agency (CBSA) for further processing. BBSA will interview and fingerprint them, and all the necessary paperwork will be filled out.
And then, almost shockingly, the families or individuals are just told to go on their way! (Unless there was a problem with the identification process, or if they are considered to pose a threat, or if it is thought there is a risk they may not turn up for their refugee hearing for some reason.) They can stay with family or friends or in an official shelter until their refugee hearing date.
And that's it! They are in Canada, and the regular refugee process takes place, and (usually) we have one more grateful immigrant in our population, and America has one fewer frightened transient.