The answer to your question is no! Ei is an elimination reaction which doesn't doesn't undergo any carbocation rearrangements and also it does not depend on carbocation stability, the carbocation rearrangement generally occurs in acid catalysed dehydration. pyrolytic elimination reactions proceed through a cyclic transition state, this is a beta elimination with syn stereoselectivity.
This kind of elimination occurs in:
1)esters when they are heated to 500 degrees also called pyrolysis of esters.
2)tertiary amine oxides when heated till 100 degrees also called cope elimination.
3)xanthates when heated to 200 degrees also called pyrolysis of xanthates.
All of the above reactions yield alkenes through elimination mechanism via a cyclic transition state.
Here's a quick mechanism for these reactions:
2)pyrolysis of ester
3)pyrolysis of xanthates(Chugaev reaction)
these are the mechanisms and its important remember that the alkene is the major product and alkenes are formed because of internal elimination reaction, this internal elimination is favored in above cases because it forms a cyclic transition state which is very stable.
These cope reaction in particular is very useful as it proceeds through 5 membered ring transition state which needs relatively low temperature.
but for Chugaev reaction and pyrolysis of esters the transition state is a 6 membered ring which needs more temperature than cope reaction.
I hope this helps clarify your doubt!!