Within the UK

ONS use GP Patient records to update their Migration estimates of within UK moves.  They also use Student Address data from the Higher Education Statistics Agency to adjust the estimates.

Matrix of Moves between Local Authorities in England & Wales (Northern Ireland and Scotland shown as entire countries only)
Mid 2019 (LA & Regions Square Matrix of Moves) (2019 & 2020 LA Geog)
Mid 2018 (LA & Regions Square Matrix of Moves)
Mid 2017 (LA & Regions Square Matrix of Moves)
Mid 2002 to 2016 (Moves between between regions (including the countries of Wales, Scotland and Northern Ireland) 

Detailed Migration figures are published annually and since 2011, are in the form of single year estimates by gender by Local Authority.

Mid 2019 Part 1 and Part 2 (2020 LA Geog)
Mid 2019 Part 1 and Part 2 (2019 LA Geog)
Mid 2019 Parts 1 and 2 on a single zip file (2018 LA Geog)

Mid 2018 Part 1 and Part 2
Mid 2017 Part 1 and Part 2
If you are interested in a copy of the full database with all the data from 2012 to 2019 (8 million+ records), get in touch &  if you would like the figures aggregated for your Local Authority/Authorities, I can supply an Excel spreadsheet that will allow you to cross-tabulate age groups and gender for the years from Mid 2012 to Mid 2019.

International Migration

ONS publish a Local Authority Migration Indicator (LAMI) Tool that shows trends in admin data sources linked to international migration e.g. NINO, Births to overseas mother, Flag 4 (GP registrations for new overseas patients), Short Term and Long Term Estimates, turnover and rates.  It covers the years from 2008 to 2019. See LAMI Migration Tool.
NRS Publish a Scotland Versions available here

UK The latest quarterly estimates (Year to March 2020) for International Migration to and from the UK can be found here
NRS for a Scotland Specific Report, it’s here
NISRA for a Northern Ireland Specific Report it’s here

Short Term International Migration

Figures for Local Authorities for Mid 2018 (published 21/5/20) available here.