You are generally alright, but consider the following.
A queen excluder below the brood for a very few days can be used to prevent her leaving.
The usual thing to do is to hive them in the evening - they are less likely to depart later in the day.
Casts are usually more prone to absconding than primes (given enough space).
If they swarmed very recently, they need not be fed for the first 3 days. That way any disease pathogens will be assimilated, not stored. Not usually a problem, but better safer than sorry later.
Nobody wants to lose a swarm after hiving it, but it does happen. They have departed from an occupied bait hive before now.
Local can mean 'close by' or the 'general area'. Swarms moved out of their locale are less likely to abscond (not scouted other alternatives).
A feed will help them to draw comb quickly, but needs only to be offered until they can start bringing in their own nectar. Swarms have managed without extra feed for millions of years, but they were not interfered with, by humans, until very recent times! So watch the weather! Conditions can change quite quickly and we are not yet certain of no prolonged cold/wet weather, when they might need extra feed tob avoid starvation.
If a cast, don't leave any queen excluder on for too long - she will need to get out to mate shortly.
Regards, RAB