All of the above is correct. You can transfer your ISA to another provider.
However, one factor to consider is that there will almost certainly be a delay between your funds leaving the first account and hitting the second. During this time you will suffer loss of interest for a period and this needs to be factored in to your calculations.
I recently transferred my cash ISA from a small building society to the Halifax. They were offering 6.4% fixed for four years. (I believe this has now gone down to 6.0%). The transfer took over a month to finally complete, although I lost interest for only about three weeks, so it was still a good deal.
There have been a few articles in the money press recently about this. The process is not supposed to take more than 28 days but there have been some reports of transfers taking up to three months. There has been no satisfactory explanation from the banks, except to highlight the problem that because of government rules surrounding the issue, the transfers have to be accomplished by banks sending cheques to each other instead of using the CHAPS system. This does not explain all of the delay, however. The real reason may be more sinister as, after all, the money is somewhere all the time the transfer is being processed.
It should not be a big deal but you should realise that if you lose all interest at (say) 5% for one month, it is the equivalent to your annual rate being reduced by more than 0.4% for that year.