After you’ve met Peebee and recruited her to your Pathfinding efforts, go talk to her in the escape pod section of the Tempest, and she will refer to needing help with a research project, though she’ll leave it at that for now.
Once you’ve reached the planet Voeld, Peebee will send you an email asking specifically to retrieve some rem-tech in the far east region of the planet.
Head out there and complete the glyph puzzle to collect the artifacts, then return them to Peebee back on the Tempest. She’ll thank you and ask that you meet in her apartment on the Nexus.
At her apartment, Peebee will show you the results of her project, but you’ll have to wait for her to research and learn more about the Remnant before you can finish the final part of the mission.
Eventually, once you’ve nearly completed Hunting The Archon as part of the main story, Peebee will request to see you back on the Tempest and ask for your help in finding three last pieces of Remnant tech, each of which is found on Eos, Havarl and Kadara.
Bring this back to her and the core loyalty mission itself will begin. Initiating this will send you into a off-ship mission featuring combat, so make sure to prepare yourself accordingly.
Once you’ve made it through the mission, and made your decision regarding the fate of Kalinda, Peebee will be completely loyal to you and your cause.
Drack’s loyalty mission can be started as soon as Ryder has recruited him, which will be after you’ve completed A Better Beginning on Eos. Go speak with Drack on the lower deck of the Tempest, and he’ll ask you to look into the shady dealings of a Nexus officer named Spender.
The following task is then a case of wandering around the Nexus, gathering information and talking to people to learn more about Spender’s dealings in the fallout of the Krogan clan during the Nexus mutiny negotiations.
You won’t be able to access the next part of the mission until after you’ve completed A Trail of Hope, which awards you access to the planet Kadara. Once you’ve landed on Kadara, the mission will guide you to a building where you’ll discover more information for your paper trail, after which you’ll need to head back to the Nexus to discuss matters with some key individuals.
From there, head back onto the Tempest and talk to Drack, who will ask you to travel to Elaadan with him for the main and final part of his loyalty mission, in which you must seize an important piece of cargo containing a Krogan seed vault.
The following section on Elaadan is a combat-heavy sequence, but following through on this leads to the return of the cargo. With that, you can return to the Nexus to deliver your verdict on what becomes of Spencer, and the mission finishes up.
Congratulations, you've completed the Loyalty Missions for all six members of your crew.