Why does influenza sometimes cause GI symptoms?
Article: How influenza virus infection might lead to gastrointestinal symptoms Source study: Respiratory influenza virus infection induces intestinal immune injury via microbiota-mediated Th17 cell-dependent inflammation.

The idea is that Influenza virus infection originates in the respiratory tract. In response, the body produces these CCR9+CD4+ Th cells denoted Th17 in the lungs. Gut epithelial cells produce CCL25, a chemokine made to attract cells expressing CCR9 chemokine receptor. Th17 cells are a subset of T-helper cells that produce interleukin-17, more about that here. So what they found, is that the Th17 cells produced by influenza infection in the lungs migrate to the gut through this CCR9/CCL25 interaction, and they secrete cytokines that are meant to defend against infection. Of course, mucosal immunity at the gut epithelium is always active, because there are commensal bacteria there that out-compete pathogens among other roles.

Down in a section titled "Lung-derived CD4+ T cells destroy microbiota homeostasis and promote resident Th17 cell polarization" their conclusion is that IFN-gamma secreted by the lung-derived Th17 cells destroys the homeostasis of the gut microbiota, and cause other Th17 cells to become polarized in a way that they end up producing the specific set of cytokines causing the trouble in the first place. You also have to understand that the other cytokines the Th17 cells are secreting are causing tissue damage, leading to symptoms of intestinal disease concomitantly with what we've been discussing above.

I think that the process is just recently being illuminated so it may not be completely understood, either. 


  How are the lung-derived Th17 cells getting to the gut? 


Generally, leukocytes in the lymphatics enter the blood through the left subclavian vein. Once there, they traffic to vascular endothelium expressing chemokine attractant molecules and utilize a number of adhesion molecules in concert to extravasate into the tissue expressing the attractant. This could mean multiple tissues given the right circumstances. The important thing to remember when i mentioned mucosal immunity is lots of effector cells are already there due to the presence of microbiota at all times, unlike the systemic response where effectors must be generated.

Here's a figure about extravasation, and another about mucosal immunity, and if you look here you can see that the Th17 lineage is under investigation due to it's inflammatory characteristics in autoimmune disease.