Bitcoin puts a cap of 2,400 transactions on each of their blocks, but not all blocks have this many. People broadcast transactions to miners and miners put transactions into blocks and distribute them.
Since, blockchains should be immutable, so you shouldn't append a new transaction onto a block's list. Also, this would change the value of the hash you need to find to validate the block.
It is up to you how many transactions you want per block, one or multiple. What I would do is gather the non-chained transactions, put them into a block and append that new block to the chain.