It is usual, for aircraft being transferred to new owners/countries, that the "Logs" for the whole life of the aircraft are handed over. This may or may not include the original or previous engines as, if they have been sold or exchanged, they may well be on another aircraft. References to previous engines/components will be logged in the airframe log too (or they should be!)
In addition to AD's, Mods, etc. Installed Life-Limited component Forms 1 and perhaps Form 1 evidence of component changes for the previous 2-4 years WILL be required by the recipients owner/national register. The buyer is entitled to the originals of any or all documents they require, the seller normally retains copies of the docs handed over. The buyer may also require evidence of all the last AMP actions and a forecast of when they are next due. Whether they still have the aircraft or not, most aircraft owners tend to retain copies of all their maintenance/operational/load records in case there is some need to prove/disprove previous actions.
In any case, "Owners/Operators" are mandated (under part M) to retain records in their keep until 3 years after the permanent withdrawal from service of that aircraft/engine/component. Whether it be in their current fleet or not. Maintenance companies may dispose of their records after the three year period already stated.
Old aircraft may have "Gaps" in their history (most DC piston aircraft did not record years of their early history - there was no need to then!) The trick is knowing whether it is an important gap or not...