KD # 1 was pretty flexible, I'd show up at his place, he'd pop into the WC and pop out with a sample in an 8 oz canning jar (glass) for me, I'd drive home (or to my office, or to a spot out in the woods, or whatevs, and insem).
I would just drive with the jar between my thighs. One trick we found useful, I took off with the sperm in the jar and left him a jar for next time, so if he wanted to start early, it would be good to go.
KD # 2 - he checks in with me about timing, which we have a good sense of because of the meds/ultrasound/trigger shot routine, actually, we usually have an appointment booked with someone to do an IUI, and I book those the day before, so he either produces at the midwife's office (or the REs office next month) or we have a system where I set a time with him, like my IUI appt is at 1:00, it's a 25 minute drive, so I tell him "I'll be in your driveway at 12:15, so no earlier, but I'm happy to wait" and then he just ducks out to the car with the sample next to his body, gives me the jar, and off I go.
He did tell me he prefers the mason jar to the specimen cup, since the opening is bigger, and the midwife doesn't care but the OB and RE prefer the specimen cup since it seems more medical and official and what they're used to.
Hope that helps! You get pretty good at driving around snuggling a jar if you're at this long enough. I don't worry too much about the time elapsed, within an hour is preferable, I often stop for drive-thru food or called-ahead-take out on my way home, I drive half an hour, let my dogs out to pee, and then get everything settled. I like to watch bad TV and eat post-insem, so I snuggle up with hips elevated, laptop set up for streaming a movie. I swear if I had made a baby with KD # 1, Iwould have named hir Wendy in honour of all the freakin' burgers I ate while insemming.