SQL Server - Export large table without primary key
Assuming you don't have updates or deletes against source table you can try the following :
1. Make a copy of existing table using CTAS syntax (for SQLServer it's SELECT * into source_table_copy FROM source_table). Such operation is very fast even for huge tables.
2. Add after insert trigger on source_table that copies new record[s] to source_table_copy.
3. Now when all new records in source_table go to source_table_copy as well, and you can move data from copied table to Mysql in batches. 
For instance, if you have a link between 2 servers, everything can be done within the body of TSQL stored procedure.
E.g. a piece of code that move up to 20 records to new server might look like

 --declare table variable to keep deleted records until they delivered to target host 
  BEGIN TRANSACTION;
  DELETE TOP (20) FROM source_table_copy OUTPUT DELETED.* INTO @Table_Var;

  --insert data into linked server , or to csv file
  COMMIT; 


It's also possible to use CURSOR to read data and then delete with where current of clause. 

**Ideally you need to prevent applications from inserting data into source_table during step 1. If it's absolutely impossible , I'll go with an after insert trigger which is added right before step 1 and removed right after it's done which copies data to some other table I can later merge with source_table_copy.