How to deal with leftover work after changing employer? In the academic world, how is work generally handled that has not been finished when an official contract ends? This is, I imagine, more the rule than the exception, for example when papers were submitted shortly before the end of the contract.

Is it common practice to "bill" (i.e. work within the work hours of) the new employer? That seemed ridiculous to me until I thought that the same situation of unfinished work will likely arise with the current employer also. Quite a number of academics, including me, are on short term contracts (&lt; 1 year). Recently I had to sink about 1 week of work time into such a former unfinished work which of course delays my current project. 

Is it expected that I work on it in my free time as a commitment to my former employer and see it as a personal mission to finish this work? Should I request a compensation for it?
This is maybe handled differently in different fields, but in any case you should inform you new advisor that you have to finish work from your previous position and discuss how this should be handled. Here are some models that I know:


Often it is expected, that postdoc do some independent work, so using this time to finish work with another workgroup is totally fine is this case.
In case your new position does not allocate time to do such work, you should inform your previous advisor. You may offer to work in your free (i.e. "non-work") time and ask for compensation (e.g. using a special contract - in German this could be a "Werkvertrag" but I don't know an English translation).
Your new advisor may get "on board" of your started project, so that your work on the project also falls under your new contract.