The UK and the European Union have agreed an extension to the Brexit deadline until 31 January 2020. This means that the UK will not be leaving the EU on Thursday 31st October 2019.
It is possible that the UK could leave the EU before 31 January if the UK parliament agrees the Withdrawal Bill, and so leave with an agreed deal in place.
During the extension period up to 31 January, you should continue to move equines as you do now to the EU, using all the current processes and systems in place. There will be no additional checks taking place during this time either in the UK or the EU.
However, it is important that you continue to be aware of what it is that you need to do to get ready for all potential Brexit scenarios during this period.
All of the relevant guidance is at https://www.gov.uk/guidance/export-horses-and-ponies-special-rules