Apollo 23 restrictions on time travel
Time travel stories must deal with the issue of how space and time are related in the imagined time travel process. I imagine that time travel is fundamentally a problem of information transfer through dimensions that are not recognized by current science.
Space and Time
To travel back in time to a particular location (such as a specific place on Earth), the "time travel transmitter" must be reasonably close to Earth. For this story, time travel into the past on Earth is accomplished by using a "time travel transmitter" that is inside the Hektor asteroid. Note: it is also possible for a second "time travel transmitter" to allow time travel into the past of the Hektor asteroid. Jack's clone is sent to Hektor using a transmitter on Patroclus.
Matter and Energy
Due to the physical processes involved in altering timelines, there are natural restrictions on time travel. For example, the energy demands of time travel increase exponentially with the amount of matter that is introduced or removed from a timeline during a time travel event.
To work around these energy limitations, a common strategy used by the Huaoshy is to use their time travel technology to obtain a cell from a person, grow a clone, and then substitute the clone for the original person.
The larger the mass shifted during a time travel event, the greater disruption in the ability to again use time travel near that point in time. If there is a close match in the mass transferred back and forward through time, then the disruption will be minimal and it will remain possible to travel through time to another point only a few decades from the other time travel event.
Time travel is made possible by sedrons, a type of fermion not yet known to our science, but which has long been used by the Huaoshy for time travel. Any technology that makes use of sedrons involves information transfer through "compact" dimensions. Such information transfer invariably results in Information Interference. Using sedronic technology near Earth can interfere with attempts to use time travel on Earth. For this reason, the Huaoshy are careful to restrict the use of sedrons near planets such as Earth where they may need to make use of time travel.
In the case of removing Jack from Earth in 1980, initially the mass of Jack's body was replaced by inanimate matter (see diagram). However, it was later learned that Earth humans had in their possession an ancient artifact from when the Olmec Reality was terminated. Jack was cloned and the clone was substituted for Jack. This was accomplished by taking cells from Jack into the future to create the clone, then sending the clone back in time to the base on Hektor.
In the case of sending Sakir back to 1969 to take the place of Jill Lyons, another "triple switch" is performed with Sakir going from 2053 to 1969, Jill from 1969 to 2027 and some inanimate balance mass from 2027 to 2053 (see diagram). This is actually accomplished in two steps. First, the act of removing Jill from Earth in 1969 makes it possible for her clone (Sakir) to be created in 2027 and sent back to Earth from 2053. To avoid unwanted resonances during this "triple switch", a second "time travel transmitter" located on the asteroid Patroclus at Jupiter's trailing Lagrangian point is used in synchrony with the transmitter on Hektor.