"The most significant implication of UK withdrawal from the EU’s Dublin System – which determines responsibility for asylum applications – would be the loss of a safe, legal route for the reunion of separated refugee families in Europe, the House of Lords EU Home Affairs Committee’s report Brexit: refugee protection and asylum policy has found."
See: Implications of Brexit for asylum policy highlighted in new report (Irish Legal News, link):
“In a no-deal Brexit scenario, refugees could be left in legal limbo, facing months of delays and additional distress, while a new framework to allow them to reunite with their families is negotiated.
The committee urges the UK and the EU to honour the right of refugees to family reunion by agreeing a temporary extension of current family unification arrangements in the event of no-deal.”
See the report: Brexit: refugee protection and asylum policy (pdf)