MySQL 8.0

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CREATE TABLE your_table (
  UNIQUE_ID INT,
  LOCAL_ID INT,
  GROUP_ID INT,
  LOCAL_STAY_ID INT,
  ORDER_IN_LOCAL_ID INT
);

INSERT INTO your_table (UNIQUE_ID, LOCAL_ID, GROUP_ID, LOCAL_STAY_ID, ORDER_IN_LOCAL_ID) VALUES
  (1,  808, 1,  808, 1),
  (2, NULL, 1,  909, 2),
  (3,  676, 7,  676, 1),
  (4,  674, 8,  674, 1),
  (5,  352, 9,  352, 1),
  (6, NULL, 9,  134, 2),
  (7,  232, 11, 232, 1),
  (8, NULL, 11, 431, 2),
  (9, NULL, 11, 323, 3),
  (10, NULL, 11, 567, 4),
  (11, 800, 98, 800, 1),
  (11, NULL, 98, 786, 1),
  (11, NULL, 98, 345, 1);

Records: 13  Duplicates: 0  Warnings: 0


CREATE TEMPORARY TABLE temp_table AS
WITH RECURSIVE fill_nulls AS (
  SELECT 
    UNIQUE_ID,
    CASE WHEN ORDER_IN_LOCAL_ID = 1 THEN LOCAL_ID ELSE NULL END AS LOCAL_ID
  FROM your_table
  WHERE ORDER_IN_LOCAL_ID = 1

  UNION ALL

  SELECT
    t.UNIQUE_ID,
    CASE WHEN t.ORDER_IN_LOCAL_ID = 1 THEN t.LOCAL_ID ELSE fn.LOCAL_ID END
  FROM your_table t
  JOIN fill_nulls fn ON t.UNIQUE_ID = fn.UNIQUE_ID + 1
)
SELECT * FROM fill_nulls;

Records: 53  Duplicates: 0  Warnings: 0


UPDATE your_table AS orig
JOIN temp_table AS temp
ON orig.UNIQUE_ID = temp.UNIQUE_ID
SET orig.LOCAL_ID = temp.LOCAL_ID
WHERE orig.LOCAL_ID IS NULL;


Rows matched: 7  Changed: 7  Warnings: 0


SELECT * FROM your_table


UNIQUE_ID
LOCAL_ID
GROUP_ID
LOCAL_STAY_ID
ORDER_IN_LOCAL_ID


1
808
1
808
1

2
808
1
909
2

3
676
7
676
1

4
674
8
674
1

5
352
9
352
1

6
352
9
134
2

7
232
11
232
1

8
232
11
431
2

9
232
11
323
3

10
232
11
567
4

11
800
98
800
1

11
800
98
786
1

11
800
98
345
1

UNIQUE_ID	LOCAL_ID	GROUP_ID	LOCAL_STAY_ID	ORDER_IN_LOCAL_ID
1	808	1	808	1
2	808	1	909	2
3	676	7	676	1
4	674	8	674	1
5	352	9	352	1
6	352	9	134	2
7	232	11	232	1
8	232	11	431	2
9	232	11	323	3
10	232	11	567	4
11	800	98	800	1
11	800	98	786	1
11	800	98	345	1

UNIQUE_ID	LOCAL_ID	GROUP_ID	LOCAL_STAY_ID	ORDER_IN_LOCAL_ID
1	808	1	808	1
2	808	1	909	2
3	676	7	676	1
4	674	8	674	1
5	352	9	352	1
6	352	9	134	2
7	232	11	232	1
8	232	11	431	2
9	232	11	323	3
10	232	11	567	4
11	800	98	800	1
11	800	98	786	1
11	800	98	345	1

UNIQUE_ID	LOCAL_ID	GROUP_ID	LOCAL_STAY_ID	ORDER_IN_LOCAL_ID
1	808	1	808	1
2	808	1	909	2
3	676	7	676	1
4	674	8	674	1
5	352	9	352	1
6	352	9	134	2
7	232	11	232	1
8	232	11	431	2
9	232	11	323	3
10	232	11	567	4
11	800	98	800	1
11	800	98	786	1
11	800	98	345	1