Does the movement stop if the remote control goes out of range?
Yes, this is the expected general behaviour: when the receiver does not get a valid signal it releases its relay outputs and the movement stops. As you approach the range limit the response first becomes intermittent, then breaks off completely. The ranges declared for AXI sets are open-field values: SF series ~100 m, ST series ~150 m.
Going out of range produces the same result as a flat battery: the transmitter's broadcast does not reach the receiver, the receiver sees no valid command and releases its outputs. The critical point is that this transition is not a sharp boundary. As the signal level drops you first get occasional packet loss, then visible delay, and finally a complete dropout. In this intermediate zone the operator gets the feeling that "the remote is sticking".
For this reason, do not lay out the working area at the edge of the declared range. The ~100 m (SF100/SF200) and ~150 m (ST100/ST200) values stated for AXI sets are for open field; concrete walls, steel structures, stacked containers, and interference from inverters and welding machines reduce this value noticeably. As a practical rule, keep the actual working radius clearly below the declared open-field range; determine the real limit with a site test after installation.
Movement stopping at the range limit is not a fault but the intended safety behaviour; the operator should not be commanding the machine when they are too far away to see it. Conversely, the movement stopping does not remove the need for a hard-wired emergency stop on site. Confirm your set's loss-of-range behaviour from the user manual.
The full topic: 433 MHz or 2.4 GHz?