Will the local and central E-Invoices get generated automatically in BUSY or do I need to do something specific?

Yes, Local and Central E-invoices can be generated from BUSY automatically and to configure it in BUSY, you will follow the given steps : 
 Step 1: Go to Administration menu then click on Configuration option. 
 Step 2: Now click on Features and options then click on GST/VAT button.
 Step 3: Now Enable the E-Invoice required option and click on configure button.
 Step 4: Mention the From Place and pincode on then e-invoice configuration page.
 Step 5: Then enable or disable the next option according to your need and click on ok button to save it.
 Step 6: Then click on GSP configuration and You will see options of enter Webtel and pinnacle ID'S.
 Step 7: Open E-Invoice Portal and click on registration option and enable e-invoicing then Click on left side e-Waybill portal using your existing credentials.
 Step 8. Go to Registration for GSP and Authenticate yourself with OTP then Add New user.
 Step 9. Choose GSP Name as Webtel Electrosoft Pvt Ltd or pinnacle finserv advisors private limited then Enter the appropriate Username and Password and add the user.
 Step 10. Enter the same username along with prefix and password in the credentials table under GST/VAT GSP configuration in BUSY.
 Step 11. Go to Voucher series Configuration and select voucher type and it's series in which you wants to generate E-Invoice.
 Step 12. On the Right side Click on voucher configuration and Enable Transport window and E-Invoice option on right side.
 Step 13. Now add voucher and it's saving time you will see the window,Do you want to generate E-Invoice and then click on it.
 Step 14. You will see transport window fill it accordingly then select Automatic and Check possible errors and resolve it accordingly.
 Step 15. Press F2 and save this window and generate E-Invoice and print E-Invoice.