Ukranien citizens from the Ukraine need not apply for asylum in the European Union under the Temporary Protection Directive of the European Commission. For nationals of other nations fleeing from Ukraine you are able to pass through borders into the EU but there are specific requirements to establish either refugee status or to transfer back to your home country. You can bring your pets and personal belongings.
[UK] Handbook Germany for advice on various agencies and information available: https://handbookgermany.de/de/ukraine-info/ua.html
[UK] Information from Berlin: https://www.berlin.de/ukraine/uk/
[EN] Border Crossing Information: https://www.mapotic.com/ukraine-border-crossings