1. Dump structure of the big database.
2. Open the sql file and cut the constraints.
3.  Import the structure to another database.
4. Run this code:

/**
 * Sync from live database
 */
ini_set('memory_limit','512M');

$limit      = 100000;
$time       = time();
$sleep      = 1;
$sleepEvery = 60;

$link = new mysqli(
    "__HOST__",
    "__USER__",
    "__PASSWORD__",
    "__DATABASE__"
);
$link->set_charset("utf8");
  
$importLink = new mysqli(
    "__HOST__",
    "__USER__",
    "__PASSWORD__",
    "__DATABASE__"
);
$importLink->set_charset("utf8");
$result = $link->query('SHOW TABLES');


$tables = [];
while($row = mysqli_fetch_assoc($result)) {
    /**
     * Filter the tables if you want
     */
    $table = $row[key($row)];
    $tables[] = $table;
}

foreach ($tables as $table)
{
    $offset = 0;
    $hasRows = true;
    while ($hasRows) {
        echo "Excecuting SELECT * FROM `$table` LIMIT $limit OFFSET $offset" . PHP_EOL;
        flush();
        $result = $link->query("SELECT * FROM `$table` LIMIT $limit OFFSET $offset");
        if (!$result) {
            $hasRows = false;
            break;
        }
        
        $rows = [];
        while($row = mysqli_fetch_assoc($result)) {
           $rows[] = $row;
        }
        
        if (!empty($rows)) {
            dumpToDatabase($table, $rows);
        }
        
        $offset += $limit;
        takeANap();
        if (count($rows) < $limit) {
            $hasRows = false;
            break;
        }
    }
}


function takeANap()
{
    global $time, $sleep, $sleepEvery;
    if ($time + $sleepEvery <= time()) {
        $time = time();
        echo 'Sleep for ' . $sleep . PHP_EOL;
        flush();
        sleep($sleep);
    }
    
}

function dumpToDatabase($table, $rows)
{
    global $importLink;
    echo 'begin_transaction' . PHP_EOL;
    $importLink->begin_transaction();
    $inserts = [];
    foreach  ($rows as $row) {
        $values = [];
        foreach ($row as $id => $value) {
            $values[] = "'".mysqli_real_escape_string($importLink, $value)."'";
        }
        
        $inserts[] = sprintf(
            '(%s)',
            implode(',', $values)
        );
        
        if (count($inserts) >= 1000) {
            $str = sprintf(
                'INSERT INTO `%s` (%s) VALUES %s;',
                $table,
                implode(',', array_keys($row)),
                implode(',', $inserts)
            );

            $importLink->query($str);
            echo mysqli_error($importLink) . PHP_EOL;
            $inserts = [];
        }
    }
    
    if (!empty($inserts)) {
        $str = sprintf(
            'INSERT INTO `%s` (%s) VALUES %s;',
            $table,
            implode(',', array_keys($row)),
            implode(',', $inserts)
        );

        $importLink->query($str);
    }
    echo 'commit' . PHP_EOL;
    $importLink->commit();
}


die('END');

5. Add the constrains from step 2.

Easy as that.