I don't know if it still applies in this update, but in previous versions a dothraki faction invades essos a few hundred days in. After that you might be able to set your faction culture to dothraki, if you have an independent kingdom, but I've never tried this. Only other thing I could recommend would be to take/rescue prisoners and recruit them. It'd be difficult to keep an entire army's numbers up this way unless your surgery skill very high... And even then..