What does the Overrun Kingdom casus belli cost?

piety and prestige; the amount depends on the titles, claims and distance, worked out when you declare.

Casus belli · Invasions

Overrun Kingdom

Overrun Kingdom is a casus belli in Crusader Kings III (invasions). If the attacker wins: The attacker seizes all titles within the defender's realm belonging De Jure to the target title. One County becomes their Capital, a number convert to their Culture and Faith, and they becomes. If the defender wins: The defender keeps their titles. Declaring it costs piety and prestige. The war goal is worth 80% war score. Patch 1.20.0.3, from game/common/casus_belli_types/09_mpo_wars.txt.

Quick answers (3)

What do you get for winning a Overrun Kingdom war in CK3?: The attacker seizes all titles within the defender's realm belonging De Jure to the target title. One County becomes their Capital, a number convert to their Culture and Faith, and they becomes (patch 1.20.0.3).
What happens on a white peace?: The defender keeps their titles.
What does the Overrun Kingdom casus belli cost?: piety and prestige; the amount depends on the titles, claims and distance, worked out when you declare.

How the war ends

Attacker wins

The attacker seizes all titles within the defender's realm belonging De Jure to the target title. One County becomes their Capital, a number convert to their Culture and Faith, and they becomes

White peace

The defender keeps their titles.

Defender wins

The defender keeps their titles.

If the war becomes invalid

The … has ended: the defender no longer … any of the claimed Counties.

Who can declare it

Target must have at least one non-Nomadic County

Every invasions war (the group's allowed_for_character) (7 lines of game script)

herders_and_tributary_constraints = yes
pam_personal_pacifism_offensive_war_ban_trigger = yes
tgp_dynastic_cycle_offensive_wars_ban_trigger = yes
tgp_war_ban_all_ministers_trigger = yes
NOT = {
	government_has_flag = government_is_theocracy
}

As written in game/common/casus_belli_groups/00_casus_belli_groups.txt.

Attacker (allowed_for_character) (40 lines of game script)

is_independent_ruler = yes
government_has_flag = government_is_nomadic
trigger_if = {
	limit = {
		is_ai = yes
	}
	OR = {
		is_landed = no
		capital_county = {
			any_county_situation_sub_region = {
				situation_sub_region_has_county = prev
				sub_region_current_phase = situation_steppe_havsarsan_zud_season
			}
		}
		AND = {
			ai_boldness >= 50
			capital_county = {
				has_bad_season_nomadic_capital_root_trigger = yes
			}
		}
		mpo_nomad_invasion_war_ai_trigger = {
			NOMAD = root
		}
	}
	trigger_if = {
		limit = {
			NOT = {
				capital_county = {
					any_county_situation_sub_region = {
						situation_sub_region_has_county = prev
						sub_region_current_phase = situation_steppe_havsarsan_zud_season
					}
				}
			}
		}
		NOT = {
			sub_realm_size > 40
		}
	}
}

As written in game/common/casus_belli_types/09_mpo_wars.txt.

When it can start (valid_to_start) (52 lines of game script)

scope:target = {
	trigger_if = {
		limit = {
			scope:attacker = {
				is_ai = yes
			}
		}
		any_in_de_jure_hierarchy = {
			continue = {
				tier > tier_county
			}
			count >= 6
			tier = tier_county
			title_province = {
				NOR = {
					has_holding_type = nomad_holding
					has_holding_type = herder_holding
				}
			}
			holder ?= {
				OR = {
					this = scope:defender
					is_vassal_or_below_of = scope:defender
				}
			}
		}
		scope:defender = {
			NOT = {
				government_has_flag = government_is_nomadic
			}
			NOT = {
				any_character_war = {
					using_cb = mpo_nomad_invasion_cb
				}
			}
		}
	}
	custom_tooltip = {
		text = at_least_one_non_nomad_county_tt
		any_de_jure_county = {
			holder.top_liege = scope:defender
			NOT = {
				title_province = {
					OR = {
						has_holding_type = nomad_holding
						has_holding_type = herder_holding
					}
				}
			}
		}
	}
}

As written in game/common/casus_belli_types/09_mpo_wars.txt.

Cost to declare

Costs piety and prestige. The amount is worked out when you declare (by the titles targeted, claims, distance and more), so the script is shown as written:

Cost (8 lines of game script)

piety = {
	value = 0
	add = common_cb_impious_piety_cost
	multiply = 1.5
}
prestige = {
	value = nomad_invasion_cb_prestige_cost
}

As written in game/common/casus_belli_types/09_mpo_wars.txt.

Source: game/common/casus_belli_types/09_mpo_wars.txt and the English localisation, Crusader Kings III 1.20.0.3. “The attacker”, “the defender” stand where the game names the character; “…” marks other words it fills in during play.