You have it backwards: it affects LIFO (and average cost) but not FIFO.
Conceptually, in a periodic inventory system you use all of the goods available for the whole period (one year, for example) and determine which costs go to COGS (and, consequently, which go to ending inventory (EI)), whereas in a perpetual inventory system you use all of the goods available as of the date of each sale to determine the COGS for that sale.
Recalling that under FIFO the earliest costs go to COGS, you can see that the earliest costs are the same whether we look at individual sales on 1/15, 2/27, 4/18, 6/3, 8/22, 10/14, 11/11, and 12/23, or we wait until 12/31. The earliest costs are always the earliest costs, no matter when we look.
However, under LIFO, because the latest costs go to COGS, things are considerably different. Under a perpetual LIFO system, for the sale on 1/15, we have to assign costs from beginning inventory (BI) or purchases on or before 1/15; either way, we’re going to get some very early costs in COGS. Under a periodic LIFO system, it’s very likely that those costs will all remain in EI.
Similarly, under perpetual average cost, the early sales get average costs from the earliest inventory (BI or the earliest purchases).
Think of it this way: suppose that we have the sales dates given above and we make a final purchase of inventory on 12/28. Under periodic LIFO and average cost, at least some of the costs of that purchase will go into COGS (even though in reality we couldn’t have sold that inventory); under perpetual LIFO and average cost, none of the costs of that purchase will go into COGS (because it wasn’t available before any of the sales dates).