MySQL 8.0

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WITH yourTable AS (
    SELECT 1 AS review_id, NULL AS old_group, NULL AS new_group UNION ALL
    SELECT 2, NULL, NULL           UNION ALL
    SELECT 3, 'BATCH', 'LEVEL_1'   UNION ALL
    SELECT 4, NULL, NULL           UNION ALL
    SELECT 5, 'LEVEL_1', 'LEVEL_2' UNION ALL
    SELECT 6, 'LEVEL_2', 'LEVEL_3' UNION ALL
    SELECT 7, NULL, NULL           UNION ALL
    SELECT 8, NULL, NULL           UNION ALL
    SELECT 9, NULL, NULL
),
cte1 AS (
    SELECT t.*,
        CASE WHEN old_group IS NOT NULL AND
                  (LAG(old_group) OVER (ORDER BY review_id) IS NULL OR
                   LAG(old_group) OVER (ORDER BY review_id) <> old_group)
             THEN 1 ELSE 0 END AS old_flag,
        CASE WHEN new_group IS NOT NULL AND
                  (LAG(new_group) OVER (ORDER BY review_id) IS NULL OR
                   LAG(new_group) OVER (ORDER BY review_id) <> new_group)
             THEN 1 ELSE 0 END AS new_flag
    FROM yourTable t
),
cte2 AS (
    SELECT t.*, SUM(old_flag) OVER (ORDER BY review_id) AS old_grp,
                SUM(new_flag) OVER (ORDER BY review_id) AS new_grp
    FROM cte1 t
)

SELECT review_id, old_group, new_group,
       COALESCE(MAX(new_group) OVER (PARTITION BY new_grp),
                MAX(CASE WHEN new_grp = 1 THEN old_group END) OVER ()) AS desired_result
FROM cte2
ORDER BY review_id;

review_id	old_group	new_group	desired_result
1	null	null	BATCH
2	null	null	BATCH
3	BATCH	LEVEL_1	LEVEL_1
4	null	null	LEVEL_1
5	LEVEL_1	LEVEL_2	LEVEL_2
6	LEVEL_2	LEVEL_3	LEVEL_3
7	null	null	LEVEL_3
8	null	null	LEVEL_3
9	null	null	LEVEL_3