I started my post-military career in desk-side support and have steadily moved through the ranks throughout the years. I can agree 110% that the main problem plaguing anyone from Tier II/III is the lack of details on tickets escalated from Tier I.
When I was Tier I, it was largely due to a lack of communication and/or management not holding Tier I to a standard of work. Once I got to Tier II and beyond, I made it a point to try and have Tier I agents sit with me for a day, or I would sit with them, to ensure that there was a level of understanding of the work that happened in the "greener pasture" for both of us.
On the technical side, I worked closely with other engineers at one company and we created a VB script that would reach out and run tracert and pings and save results to a text file that would be added to a ticket. After I left, a bright young Tier I "earned her stripes" and took my place after turning all of our VB scripts into one single application written in PowerShell and compiled into an .exe
However, I will always go back to a strong belief that 99% of the issues between tiers of support can be avoided with open communication and a management "buy-in" of a team mentality across all tiers, instead of merely within them.