SLF here, but question to the experts: after having experienced such a substantial detour through unpaved area, throwing up mud and whatever - would you continue at your own power? I'd be worried about any ingested mud, gravel, ice etc and rather shut down the engines? Furthermore I'd be worried about any damage to other aircraft parts, landing gear, hydraulics that might have a substantial impact on the ability of controlling the aircraft whilst taxyiing and / or increase the damage to the aircraft by taxiing to the stand?
Yes… especially engines could be damaged. Possibly with a fire outcome.