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Ok, you're a nifty new recruit for a space marine chapter. You've passed your entrance exam without dying, and now you are headed up into the void of space for the first time (presumably in a thunderhawk or something). Waiting for you on the edge of the system is.....what? A battle barge? Most chapters only have one or two of them, and sending a ship capable of landing three companies of space marines simultaneously seems a little excessive, even for the future, currently vulnerable future of a chapter. A Space marine cruiser? Better, but still capable of deploying a full company of marines in under 20 minutes, and geared more for planetary assault than picking up hitchhikers.
I'm guessing initiates would be transported in a Gladus class frigate or somesuch (yay lexicanum), but they seem awfully small to be travelling alone.
I know some chapters recruit from their home planet, or home system like the ultramarines, and others like the Dark Angels and Space Wolves have a floating fortress or battlefleets that could probably pick up initiates as they go by, but what about the rest?

Most chapters recruit from their homeworld.
Other fleet-based chapters recruit from a few worlds their fleet visits, so recruits would be picked up by that fleet.
In all other circumstances the recruits would be picked up by whatever ship the Marines used to get to the planet they recruited on. They probably use a frigate or cruiser or whatever ship is available at the time. Perhaps the 10th company has its own cruiser?

It could just be a cargo transport or some other auxilary craft attatched to their fleet who are transporting supplies in a convoy.
That or they may train them on the 10ths vessel as said above when they get picked up after a recruitment drive.