# Recommend not using '*' but specifying the field names, 
#   to future-proof code.
my $query1 = "SELECT * FROM database1.table WHERE ( id = ? );";
my $query2 = "INSERT INTO 
  database2.table ( value1, value2 ) 
  VALUES ( ?, ? )
;";
my $sth1   = $dbh1->prepare( $query1 ) or die $dbh1->errstr;
my $sth2   = $dbh2->prepare( $query2 ) or die $dbh2->errstr;

foreach my $id ( @big_data_arrray ) {
    # Original code had an error 
    #   $sth2->execute was called in both cases
    $sth1->execute( $id ) or die $dbh1->errstr;
    while ( my @row = $sth1->fetchrow_array ) {
        $sth2->execute( @row[0 .. 1] ) or die $dbh2->errstr;
    }
    $sth1->finish;
}