So why search her only at the border then, rather than obtain a warrant to search her domicile? (Please correct me if this in fact did occur as well.)
If this was indeed an investigation passing usual legal requirements with the intent to build a case against her, I'd imagine that any responsible investigator would at the very least choose to do that as well. If they repeatedly searched her at the border (thus essentially abusing an avenue which had its legal hurdles significantly reduced for the ostensible sake of national security and being able to deal with external threats that the internal legal system had no time and opportunity to handle through regular channels) but did not choose to obtain a warrant or otherwise start a legal process that comes with the normal set of safeguards, I'd say that strongly seems to indicate they did not have enough of a case against her - or, in other words, that an investigation was perpetuated which according to traditional legal standards should have been discontinued.
Now, of course you may argue for the traditional standards/safeguards themselves to be weakened (e.g. by saying that while searches of one's home should remain subject to the same legal restrictions, certain parts of the government now also shall have the privilege to conduct some sort of "preliminary searches" which do not come with the usual safeguards to arbitary ends which might include a more formal investigation being started, at their full discretion), but this is not how I usually see this debate being framed.
The most plausible reason to me would just be the fact that it's DHS doing the searching, not FBI. Border control has very broad authority to search and question anyone coming into the US, and some office at DHS may be very interested in her but the equivalent office at FBI might not have the time or inclination to pursue it further, especially if the searches and questioning keep turning up nothing. I worked in the military in the past and have seen plenty of examples of government organizations with different but slightly overlapping missions. They tend to step on each others' toes just as often if not more often than they work together effectively.
I also have no idea if the FBI pursued it further and got a warrant. They're under no obligation to inform someone that they're under investigation. They may very well have started up a case against her that's still open, or may have investigated her and eventually dropped the case, or may not have done anything at all. In any case, they've never charged her with a crime and she's never been arrested.
I see how this being a DHS endeavour would give rise to the observations, but why would the offense in question, assuming it is one, be a homeland security rather than FBI or military police matter?
Only because border control falls under the purview of DHS. If they did find something illegal, as far as I know all they could do is either deny her entry or hand her and the evidence they collected over to the FBI and see if the DOJ wants to bring charges.
It definitely wouldn't be a matter for the military police - since she isn't a member of the military she isn't subject to the UCMJ. The only reason they would have done anything in Iraq would be because Poitras was in a war zone outside US territory. They, again, would have had to hand her over to the DOJ for prosecution.
If this was indeed an investigation passing usual legal requirements with the intent to build a case against her, I'd imagine that any responsible investigator would at the very least choose to do that as well. If they repeatedly searched her at the border (thus essentially abusing an avenue which had its legal hurdles significantly reduced for the ostensible sake of national security and being able to deal with external threats that the internal legal system had no time and opportunity to handle through regular channels) but did not choose to obtain a warrant or otherwise start a legal process that comes with the normal set of safeguards, I'd say that strongly seems to indicate they did not have enough of a case against her - or, in other words, that an investigation was perpetuated which according to traditional legal standards should have been discontinued.
Now, of course you may argue for the traditional standards/safeguards themselves to be weakened (e.g. by saying that while searches of one's home should remain subject to the same legal restrictions, certain parts of the government now also shall have the privilege to conduct some sort of "preliminary searches" which do not come with the usual safeguards to arbitary ends which might include a more formal investigation being started, at their full discretion), but this is not how I usually see this debate being framed.