As with most large scale changes, it was slow and uneven. Soldiers from the States came with their own kit. If the unit was put together in the States and shipped over in one go, soldiers tended to have the same uniform. However, not necessarily the most current. If the depot supplying them had older kit, older kit is what they got. They didn't hold a unit back until the latest kit could be sourced.
Mixing/matching was standard throughout the entire war. Soldiers in the field stuck with what they had until they wore it out for the most part. Individual replacements (which was the US system at the time) could have anything on their backs. If a unit was pulled back from the front for an extended period of time they would be more likely to be refitted with new kit vs. one that was pulled back to an immediate reserve position. Etc.
Little known fact is that Springfield rifles were still commonly issued, even in France 1944, because of a shortage of M1 Carbines. Which is another angle... generally speaking the combat troops got the newest stuff first, the rear units got them last. So you could have a Rifle Battalion where part of the force had the latest and greatest, parts of it a mix, parts of it only the old stuff.
Think about it in current terms. Much more sophisticated warehousing systems, more "in time production", and yet transitions still take years. When you think about WW2, the timeframe for change isn't so different.