I will address this from the perspective of a single member LLC reporting as a disregarded entity. That is, you would have and LLC, but still report it on a Schedule C on your personal tax return. Tax matters would be different if you were to elect S-corp status for the LLC.
Generally speaking, your income is considered earned based on where you are doing the work. If you are in a foreign country and performing the work for your US clients, that is considered foreign earned income. If you traveled to the US and performed work for the same US clients, that is considered US earned income because you are physically in the US at the time of the work. So, basically, the foreign earned income works the same for employees and for self employed persons.
However, there are more issues to consider when you are self employed. You will be subject to self employment tax (15.3% of net income for Social Security and Medicare tax) on the net income from your business. The country you are living in may also have a type of self employment tax. US tax guidelines determine that the self employment tax is paid to the country where the work is performed. When you are working on your tax return, you may need to make an adjustment to prevent the business income from being taxed for self employment tax in the US. It is also possible that you will be eligible for the foreign tax credit for taxes paid to the country you are working from.
International Social Security Agreements
The United States has entered into social security agreements with foreign countries to coordinate social security coverage and taxation of workers employed for part or all of their working careers in one of the countries. These agreements are commonly referred to as Totalization Agreements. Under these agreements, dual coverage and dual contributions (taxes) for the same work are eliminated. The agreements generally make sure that social security taxes (including self-employment tax) are paid only to one country.
https://www.irs.gov/individuals/international-taxpayers/self-employment-tax-for-businesses-abroad
