The Lone Wanderer can go about this situation in one of six ways:
She can be killed to obtain the 400 caps along with her equipment with no Karma change unless taking stuff from around her shack. She will attack the Lone Wanderer straight away if they tell her that they are there for the caps.
With a successful Speech attempt, the Lone Wanderer can convince her to give them 400 caps to tell Moriarty that she's gone for good.
She can be convinced to give over 300 caps to tell Moriarty that she's gone for good.
The Lone Wanderer can ask for no money and tell Moriarty that she's gone and couldn't be located. This will result in positive Karma, though they are still going to have to come up with 100 to 300 caps.
It's possible to simply say that she's gone without working out a deal with her, though this will result in having to come up with the money themselves.
With the Speech check, the Lone Wanderer can make her give them the 400 caps. They can then kill her to loot another 100 caps from her body. The Lone Wanderer will lose Karma if they kill her after she gives them the 400 caps.
Do not agree to tell Moriarty she is gone before talking to Moriarty, or he may spawn into an inaccessible area; see his own article for tips on fixing the problem on the PC version (console players cannot fix the problem). Alternatively, agreeing to his before speaking to Moriarty will simply remove all dialogue options with him to tell her she is gone.
Completing the Megaton part of Following in His Footsteps will also remove the possibility of completing this quest (even if you talk to Silver first about it).
If the Lone Wanderer talks to Madison Li about their dad before completing this quest, all quest dialogue with either Moriarty or Silver will disappear and Silver will offer only post-quest dialogue.
If the player has spoken to Moriarty at least once, but has not touched up on the subject of Silver, when finally speaking to Silver, the dialogue options act as if the player has indeed touched up on the subject with Moriarty and the player already knows the situation, even if it was never previously covered in dialogue or no part of the Following in His Footsteps quest was advanced beyond searching Megaton for answers.