Just take Military Control of the puppet. If you have taken Military Control of your puppet - meaning you can move its units - then you should have no problem loading them onto your transports. I and most others do it all the time.
If you expedition either the forces being moved by ship or the transports being used (includes your own expeditioned to anybody else) THEN YOU RISK CHANGING OWNERSHIP OF THE FORCES UPON LANDING.
Never expedition anything to do with transporting troops unless you want to steal the puppet's divisions (to become your own) and know very well what you are doing as the possible outcomes are endless. Simply put, when a unit starts expeditioned to country A, and is moved to country B, the expedition flag - once cleared - puts ownership of the unit to the previous country. As an accident it works every time. As deliberate stealing the AI is quite wise to seeing a repetition; and stops further units changing ownership. Then it takes clever changing to different country transports or landing at different country to continue stealing.
EXAMPLE: I just had a major disaster. Because I am low on MP, I want the UK Armor to take Africa with a landing south of Casablanca. I Military Control the UK. But if do an amphib using their units, the won landing place will become UK. That is undesirable because I repair provinces much faster than the UK. Ideally I needed a US unit in the amphib that would land first - but how does one beat the speed of ARM-VII? And I had no paratrooper on that side of the world.
So I thought I would be Mr. Wise Guy and expedition the UK Armor corps (7divisions) to the USA by landing them first at the nearby US controlled territory of Gran Canaria. That done - UK armor corps expeditioned to the US - the UK armor did its amphib, and claimed the newly landed province for good old USA. I thought I was really clever... until I noticed all the 7 ARM divisions have changed ownership to USA. Their expedition flag cancelled upon landing, and ownership went to last country they were in (USA).
Now I am paying the manpower losses for that whole African campaign instead of the UK. What a dumb disaster!
I should never have expeditioned those units and then put them on a transport. I should have used them as proper UK units to knock out the defenders, then cancelled their amphib, then did new amphib using a USA INF I had in the area (to claim the territory for USA) and then landed the British force.