CROSS/OUTER APPLY in MySQL

Starting from MySQL 8.0.14 you could use LATERAL:

A derived table now may be preceded by the LATERAL keyword to specify that it is permitted to refer to (depend on) columns of preceding tables in the same FROM clause. A derived table specified with LATERAL can occur only in a FROM clause, either in a list of tables separated with commas or in a join specification (JOIN, INNER JOIN, CROSS JOIN, LEFT [OUTER] JOIN, or RIGHT [OUTER] JOIN). Lateral derived tables make possible certain SQL operations that cannot be done with nonlateral derived tables or that require less-efficient workarounds

CROSS APPLY () <=> ,LATERAL ()
OUTER APPLY () <=> LEFT JOIN LATERAL () ON 1=1

Support for LATERAL derived tables added to MySQL 8.0.14

And in this case:

SELECT ORD.ID
    ,ORD.NAME
    ,ORD.DATE
    ,ORD_HIST.VALUE
FROM ORD,
LATERAL (
    SELECT ORD_HISTORY.VALUE
    FROM ORD_HISTORY
    WHERE ORD.ID = ORD_HISTORY.ID
        AND ORD.DATE <= ORD_HISTORY.DATE
    ORDER BY ORD_HISTORY.DATE DESC
    LIMIT 1
    ) ORD_HIST;

If correlated subquery does not return any rows, the main row from main query will be skipped. In such scenario, LEFT JOIN LATERAL should be used.

db<>fiddle demo

Your closest direct approximation is a join with a correlated sub-query as the predicate.

SELECT
   ORD.ID
  ,ORD.NAME
  ,ORD.DATE
  ,ORD_HISTORY.VALUE
FROM
  ORD
INNER JOIN
  ORD_HISTORY
    ON  ORD_HISTORY.<PRIMARY_KEY>
        =
        (SELECT ORD_HISTORY.<PRIMARY_KEY>
           FROM ORD_HISTORY
          WHERE ORD.ID = ORD_HISTORY.ID
            AND ORD.DATE <= ORD_HISTORY.DATE
       ORDER BY ORD_HISTORY.DATE DESC
          LIMIT 1
        )

In your case, however, you only need one field from the target table. This means that you are able to use the correlated sub-query directly in the SELECT statement.

SELECT
   ORD.ID
  ,ORD.NAME
  ,ORD.DATE
  ,(SELECT ORD_HISTORY.VALUE
      FROM ORD_HISTORY
     WHERE ORD.ID = ORD_HISTORY.ID
       AND ORD.DATE <= ORD_HISTORY.DATE
  ORDER BY ORD_HISTORY.DATE DESC
     LIMIT 1
   )   AS VALUE
FROM
  ORD