This type of move is becoming more common as people find they are unable to get mortgages and so rent while they save for deposits etc, by definition this type of move is better for removers as there is no delay for the completion and release of keys, however there are still potential delays.
The main one is the delay caused when the handover is done on the moving day, this can be up to 3 hours long and some agents will not allow any items moved in until they have completed the hand over.
It is always a good idea if possible to get an overlap between properties, this will allow time to get the handover done the day before moving and also if it is a big move will allow removal of goods over a longer period of time which could reduce your moving costs and will certainly reduce your stress.
By arranging an overlap and maximising the flexibility of your move you are more likely to get the remover you want.