Is it permissible to give money to restaurants that distribute expiatory payments, or must they be distributed personally? And must expiatory payments be made in one lump sum or over extended periods?

If you are certain or highly confident that the restaurants fulfill the expiations as prescribed by Islamic law, then there is no harm in giving them money to distribute the expiations on your behalf. However, if you are not certain, then you must distribute them yourself, because one's obligation is not discharged except with certainty. It is not a condition to feed the needy all at once; rather, the obligation is to complete the required number, even if it is at different times.