Normally trains from both systems through-run, and indeed rail companies will pretty much always have their own stock regardless of how short the line or who actually operates it - e.g. the 2.5 mile-long Yokohama Minatomirai line, effectively an extension of the Tokyu Toyoko line and operated by Tokyu crews, still has its own fleet of EMUs. However some classes of train may be barred from metro or suburban lines due to lacking required safety systems etc.; an example would be the Tokyo Metro Tozai line, which connects to the JR East Chuo-Sobu line at its western end and to the Toyo Rapid line at its eastern end - Tokyo Metro stock can run on all three lines, but JR East stock can't run on Toyo Rapid metals and Toyo Rapid stock can't run on JR East metals as they use different safety systems.
In most cases through-running is something that happens all-day, and indeed on some suburban lines all trains through-run. Peak-only extensions are not uncommon, though, and indeed some outer suburban stations see one through-running train a day.
An additional point worth noting about Japanese through-running is that with the exception of lines owned by companies that don't employ their own train crews at all, crews are always changed at company borders, the process usually being done in less than a minute. Crews operating onto "foreign" territory used to be somewhat more common, but it was clamped down on in the aftermath of the 1991 Shigaraki crash where a JR West through service onto the Shigaraki Kogen line collided with a train travelling the opposite direction (though I can't find anything saying that the train crew's actions caused the crash in any way, which was primarily caused by dispatchers allowing a train to enter a single-track block without confirming it was clear).
