Well, for a start the deposition of Robert of Jumièges (the first Norman Archbishop of Canterbury) was deposed by the Godwins in 1052 and replaced by Stigand, then Bishop of Winchester. This was condemned by Pope Leo IX and used as a justification to invade England by William "the Bastard". The depiction of Stigand at Harold's coronation is thus meant to destabilise his reign.

As for others, they probably stayed for the most part. Bishop Wulfstan of Worcester (died 1095) was the last English bishop after 1075 and demonstrated his allegiance to William by commanding an army that stopped the advance of Robert de Breteuil, Earl of Hereford (during the Revolt of the Earls, 1075). Even Stigand remained Archbishop of Canterbury and Bishop of Winchester until 1070, when he was formally deposed by the papal legate (the Bishop of Sion) alongside the Bishops of Elmham, Selsey, Durham and Lichfield.