Who should cover conferences expenses for research work done in a lab a researcher just left for another lab?
My perspective comes as a research group manager at a US national lab. 

I pay for conference attendance for my people - it is part of their job, and their ability to grow as a researcher depends on attending conferences. Whether they are post-docs or new staff, I hired them in no small part because they were performing work that was at least tangentially relevant to their first assignments. So, I will get them to the conference to maintain and grow their reputation and skills. The cost of registration and travel is really small compared with a loaded salary. And I at least get my institution listed as their current address at the talk.

For post-docs in particular, I tell them that 'the job of a post-doc is to get a job.' The way they will get a (permanent) job is by doing good work, publishing, and presenting at conferences. They need to be visible, have chances to network, and come back with good ideas. So what if the first conference they go to is based on older work? They need to be out there to keep developing, and it is my job to make sure they develop.

Finally, I do this at least partly because that is how I was treated as a post-doc and new staff. That first conference at each institution was based on my 'old' work, but it was at conferences and in areas that were important to my 'new' job. And, my bosses agreed. The least I can do is arrange that for others. 

I understand that there may be budget issues in other situations. However, the PIs really should consider if they are being penny-wise and pound-foolish. Good things happen at conferences, for both the post-docs and the PIs.