Here is information about crossing the border between China and Kazakhstan at the Khorgas border crossing starting from Yining Ile, China and going to Almaty, Kazakhstan.
Shared taxis and buses from Yining Ile leave from the Bus Station at the Northwest end of Jiefang Road, Map Location: 43.935947, 81.297213 .
The bus station opens sometime around 08:00 or later but the shared taxes are there earlier. The shared taxis are just private cars which leave when they are full and charge around 45 CNY per person. They drop people off at the bus station in Khorgas where the buses to Zharkent depart.
The bus station opens around 09:30 to sell tickets to Zharkent for 70 CNY. The buses are actually big vans which hold about 15 people. They also leave when they are full which could be a few hours after the stated departure time. The advantage to the Zharkent bus is that it seems to have it's own route through immigration and customs so there aren't lines with other buses or people.
In Zharkent, the bus goes to a bus station where there may or may not be buses to Almaty. There are many shared taxis waiting for people to go the three hours to Almaty for about 4000 to 5000 tenge per person.
Here is an example of the travel time breakdown:
08:00 - Left Yining/Ile bus station in shared taxi. There was one long stop for a police check outside of Khorgas.
09:00 - Arrived at Khorgas Bus Station.
09:30 - Bus Station opened.
09:45 - Nasty bus station lady finished her breakfast and started selling tickets. She said that the bus would leave at 10:00 but it leaves when it is full.
11:30 - Bus was finally full and went to the China exit at the border. The China exit took an hour because an old Kazakh couple didn't know how to fill out their departure cards and got held up.
12:30 - Arrived at Kazakh Immigration/Customs. This also took about an hour
14:00 - Arrived at Zharkent Bus Station. Took forever even though it's only about 30 kms to Zharkent from Khorgas. There were no more buses to Almaty but there were many shared taxis going for about 4000 tenge (5000 if they dropped off at a particular address). The drive to Almaty takes about 3 hours with 1 bathroom stop at a horrific desert outhouse.
17:00 (15:00 Kazakh time) - Arrived in Almaty. There is a 2 hour time difference between China (Beijing time) and Kazakhstan.