You can find a application form for mobility allowance (in finnish) on the website of The Federation of Unemployment Funds in Finland. The form is called Liikkuvuusavustushakemus.
Fill in the form and save it on your computer. After that you can sent the saved form to the fund via WebKassa -on-line-service. A copy of your employment contract need to be attached to the application. If you wish, you can also send the application by post.
The payment of the allowance starts after the beginning of your employment and the allowance is paid automatically in periods of 4 weeks.
Please remember to notify the fund of any changes in your employment.