Additionally, is there any strategy to the hunt for these items? Sure there is -- if you're hunting for both Tier 4 and Tier 2 Promotion Items, then you can skip the 4-Energy Tier 2 missions and go straight to the new 4-Energy Tier 4 missions (same mission, just lowered cost), since you will be getting at least 1 Tier 2 Promotion Items in each Tier 4 mission, and you're not spending any additional Energy. Same for Tier 5 and Tier 3 -- skip the Tier 3 missions and go straight to the Tier 5 missions, since you will always get a Tier 3 Promotion Item in the Tier 5 mission. However, if you absolutely do not need the Tier 4 or 5 item and your team is not strong enough to handle those harder missions, then just hit the Tier 2 or 3 mission. As for Tier 1 -- I would recommend hitting any higher tier mission, since they will often reward Tier 1 Promotion Items as well. No need to waste the precious 4 Energies.
Click here for Armor Promotion Item screenshots.
Click here for Force Promotion Item screenshots.
Click here for Mechanical Promotion Item screenshots.
Click here for Survival Promotion Item screenshots.
Click here for Weapon Promotion Item screenshots.
Here are screenshots of my experience (remember to check back often for updated screenshots):