The level does not increase.

Currently only way to lvl theloaner units is if you purchased packs that included something to lvl them. 

AhnaldT101 provided  all the possible sources currently available.

Everyone got five for free.

Four are awarded from Tier 4 of the Marquee.

Episode Pass awards three 

Pass Plus awards a bundle that has another five. 

Tier 1 R5-D4 $30 shard pack gives two. 

Tier 2 R5-D4 $100 shard pack gives seven.

26 total upgrades and you start at level 1 so Era Loaned Level 27 if all are applied.