Can ablution ( wudu ) can be done partly at home and completed at the mosque (especially washing legs) ?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

If the distance between the mosque and your home is so short that you can perform some of your ablution at home and complete the remaining at the mosque without the last organ you washed at home becoming dry, then it is lawful to do so. But if the last organ you washed at home dries before you get to the mosque then you cannot consider what you washed at home as the continuation of washing organs because continuation is an obligation in ablution. This is the prevailing opinion and it is the opinion of the Hanbali and Maaliki Schools of jurisprudence.

For more benefit on the continuation when performing ablution, please refer to Fatwa 139853.

Allah knows best.