What kind of antennae are used on the ISS for these links? Are they directional? Do they have to continuously move to maintain a live link - for example in the case of the live streaming HDEV views? I'm wondering if they are phased arrays which can be steered (potentially very quickly) electronically, or a more conventional gimbaled dish, a little like the picture below.
The US side of the ISS has a number of antennas to support its rather complicated comm system. Most visible are the two Ku-band High Gain Antennas which are 6 foot diameter azimuth/elevation gimbaled dish antennas mounted on the Z1 truss segment (near the center of the ISS truss). These antennas are sometimes referred to as SGANTs (Space-to-Ground ANTennas).
For most of its life, the ISS only had one SGANT. A second one was carried up on one of the later shuttle missions. This system carries the video and experiment data to the ground and would normally be gimbaling continuously to point at the selected geosynchronous comsat (i.e. TDRS).
You can see one tracking a TDRS satellite in this video (ignore the ammonia venting!)
The S-band system uses S-band Antenna Support Assemblies (SASAs) which contain a conical high gain antenna and an omnidirectional antenna. The SASAs can move in elevation and azimuth as well to point at the TDRS in use. They are located on the S1 and P1 truss sections, near the center of the truss. S-band is used for voice and data communications.
(NASA photo ISS021E033057, doesn't appear to be online at present)
There are also two Ultra High Frequency antennas on the ISS, primarily used to communicate with extravehicular activity crewmembers, formerly used to communicate with the space shuttle orbiter. They are non-gimbaling omni antennas.
My information on the Russian ISS comm system is very limited and possibly outdated but they have (or had) a system analogous to the S-band system called Regul that communicates with Russian ground stations and the former Luch satellite system. There is also a Russian UHF system used for space-to-space comm.
My small team at Dynacon in Toronto designed the software used by the gimbals to point the SGANTs (Space to Ground ANTennas); SPAR Aerospace's Montreal plant developed the SGANTs for GE (ISS comms subsystem contractor), who in turn were subcontractors to McDonnell Douglas (Work Package 3 contractor, IIRC, reporting to NASA's Johnson Space Centre). This was in the 1989-1993 time-frame.
The SGANT controller has 3 operating modes: slew, search and track. The first is open-loop: slew as fast as possible from one direction to another (basically as a TDRS (tracking and data relay satellite) sets, slew to where the next one is rising). Search is an open-loop spiral search pattern, looking for the strongest signal (i.e., the target TDRS), which takes a minute or two, after which a slew is commanded to where that signal was found. Track is a closed-loop mode, using signals from monopulse tracking sensors built into the SGANT horn to measure the pointing error wrt TDRS, which drives a feedback controller to minimize that pointing error. All pretty simple in principle, complicated by the gimbal's own internal dynamics (including friction), and the significant amount of flexibility and vibration of the ISS (including time-varying thermal distortion, which is what drives the need to search for TDRS at the start of each tracking session, as the ISS "bananas" in different directions as it heats up and cools down).
It is only used to track TDRS satellites, it does not have the slew-speed capability to track targets on Earth.