The 2011 and 2012 Airs can run very slowly if there is an issue with the battery, for example, if the wrong one is used or if there is a fault with the battery. This will cause the cursor to lag too, and is only an issue which can occur on the 2011/2012 Airs, it doesn’t happen when there is a battery fault on the newer models. I would also suggest checking the whole logic board for liquid damage, since it likely has some damage on the backside. Check the trackpad PCB and cable, in case there is corrosion (although this is unlikely to be the cause, since the cursor lagging will usually be battery or sensor related).
Check the board and hopefully you will find some corrosion to point you in the right direction